Alright, here's a brief explanation and a few comments on the Orthodox Schism, which I'll give in two sections: (1) the immediate crisis, and (2) a little bit of the deeper historical and ecclesiastical background as I see it.
The Scenario
The immediate crisis can be most easily understood by looking at things from a very basic standpoint of ecclesiastical entities and numbers. Since math is hard and exact numbers impossible to come by, these will be by design very approximate and drawn from readily-available-through-googling sources.
The Russian Orthodox Church is numerically the largest Orthodox church in the world; by usual estimates, about half of the three hundred million Eastern Orthodox in the world are under the direct jurisdiction of the Patriarchate of Moscow. It is this, rather than any ecclesiastical claim to primacy, that makes Moscow one of the most important centers of world Orthodoxy, with a great deal of sway even other Patriarchates.
The modern ROC encompasses not only the modern nation of Russia, but also the Orthodox believers in (most) former Soviet nations. One of these, by far the most important, is Ukraine.
Ukraine, like Russia, is an extraordinarily large and populous country, with about thirty million nominal Orthodox believers and a very large number of parishes, monasteries, and other ecclesiastical holdings. It is also, by most estimates, a rather more religious society than Russia, where the vast majority of the populace are nominal members of the ROC, but where actual Church attendance is the lowest of any European country (consistently under 10%, and closer to 1-2% in major cities). It is this basic issue of numbers and parishes, more than anything else, that makes the question of the Ukrainian Orthodox Church so important for Moscow. As matters stand now, roughly half of the parishes of the ROC are in Ukraine rather than Russia; and this does not including the many parishes that have left Moscow and joined other Orthodox groups in the last thirty years.
The question of Ukrainian autocephaly, then, is not at all a distant or theoretical one for the Moscow Patriarchate. Giving up Ukraine means giving up a very large proportion of believers and institutions, and all that goes with them--including a great deal of Moscow's status in the Orthodox world generally. A fully united, autocephalous Ukrainian Orthodox Church would be, immediately, the second largest Orthodox jurisdiction in the world, and a strong counterweight to Moscow. This is something that Moscow has shown itself entirely unwilling to accept.
The Patriarchate of Constantinople, on the other hand, directly presides over, in Turkey itself where it is located, one of the tiniest flocks in the world (perhaps 5,000), though it also has jurisdiction over many ethnic Greeks worldwide and most Orthodox in America and the West. Still, even with this, its numbers are a tiny fraction of Orthodox in the world (perhaps 6 million): its status in the world of Orthodoxy comes far more from the fact that it is universally regarded as the First See of all the Orthodox patriarchates and the "center" of Orthodoxy as a whole. What this means is not always very clear, but since the Eastern Orthodox churches are all descendants of the Church of the Byzantine Empire, and since Constantinople was, of course, the absolute center of that Empire and of the Church attached to it, that fact alone carries with it a great deal of prestige and influence.
The Patriarchate of Constantinople wishes to grant autocephaly to Ukraine, as it has to many other Churches formerly subject to it (including Moscow itself). Autocephaly, "having one's own head," means that the Church in question is self-governing, typically led by a Patriarch who has jurisdiction over the Church and all its dioceses and ordains its bishops.
Part of the issue is that there is no clear procedure or structural principle underlying "autocephaly" or its granting. The Byzantine Empire was an extraordinarily centralized society where Church and State were deeply entwined, with both centered around Constantinople and the office of the Emperor, who possessed religious as well as political authority. After the fall of the Byzantine Empire, however, and later with the rise of modern nationalism, Byzantine Christians became politically and ecclesiastically "balkanized," gradually divided among multiple competing political and religious structures. The practice of "autocephaly" that has become the norm in modern Orthodoxy is little more than the regularized practice of Orthodox groups, largely for political reasons, breaking off from the Constantinopolitan center and managing their own affairs instead, generally with at least nominal (even if effectively coerced) consent from Constantinople itself.
The immediate impetus for Constantinople's move in Ukraine, though, is not so much ecclesiastical order (there are nation-states whose Churches are not autocephalous, and nation-states that are) as the desire to resolve a long-standing crisis and schism in Ukrainian Orthodoxy itself.
When the Soviet Union fell and Ukraine became for the first time in modern history an independent nation, a request was made to Moscow that the Orthodox in Ukraine be immediately granted autocephaly. The request was predictably denied, but in an equally predictable response, a large group of Orthodox clergy and believers in Ukraine broke off anyway and elected their own Patriarch, Filaret Denysenko. It is very difficult (at least for me) to tell precisely how many members this Ukrainian Orthodox Church of the Kyiv Patriarchate, as it is known, has, but some polls have shown its number of self-declared members to be actually larger than the ROC in Ukraine (though the ROC has many more parishes in Ukraine, and the numbers are muddied due to the large number of Orthodox in Ukraine who had no explicitly declared allegiance at all and who probably attend, if at all, ROC parishes), and has grown in recent years due to the military conflict between Ukraine and Russia. For the last thirty years, it has, however, not been recognized by, and not in communion with, any other Orthodox Church outside of Ukraine. There is also, confusingly, a third Orthodox jurisdiction also in Ukraine, the Ukrainian Autocephalous Church, which has its roots in independent resistance among the Orthodox to Russian rule, but in the more immediate past is largely the result of distrust and hostility towards Patriarch Filaret, who had been a prominent Russian Orthodox bishop under the Soviet Union and thus is a man with strong past ties to the KGB and the Russian government. The UAOC is not in communion with any other group, including the Kyiv Patriarchate. Taken together, then, these two jurisdictions represent, both within Ukraine and for world Orthodoxy as a whole, a sizeable proportion of believers, parishes, and clergy separated from and not in communion with the rest of the Orthodox world.
Constantinople's solution to this problem is, in the abstract, relatively obvious: to officially grant Ukraine autocephaly, thus reuniting all three Orthodox groups in Ukraine into a single Church with a single Patriarch, in communion with the rest of the Orthodox world.
In reality, of course, this solution is far from simple, predominantly because Moscow (for the reasons discussed above) will never accept it, but also because of internal divisions within Ukrainian society that make reconciliation between pro-Russian and pro-nationalist factions (for obvious reasons associated strongly with the ROC and the two independent jurisdictions, respectively) very difficult. This is the basic situation leading up the events of the past few weeks and months.
The events themselves have proceeded fairly straightforwardly, though how they will proceed in the future is very open to question. Constantinople has proceeded quickly towards the process of granting the UOC autocephaly, declaring their intentions, sending exarchs to Ukraine to negotiate with the independent jurisdictions (the ROC naturally ignored them), and finally, in recent days, unilaterally declaring all the members and clergy of the UOC-Kyiv Patriarchate and the UAOC to be in communion with Constantinople and canonically regular. In the near future, one would expect, they will officially grant Ukraine autocephaly, and the Ukrainians who accept this will elect a single Patriarch (who may or may not be Filaret, as discussed above).
Moscow, on the other hand, has responded to this first with belligerent rhetoric, then with increasingly extreme canonical and ecclesiastical moves. Members of the ROC, including most prominently Metropolitan Hilarion Alfeyev, the ROC's official head of external relations, have referred to the conflict as a "war," have threatened not only to break communion with Constantinople, but also to permanently dethrone Constantinople from its position as the First See of Orthodoxy, and have compared the forthcoming schism to that between the Catholic Church and Eastern Orthodoxy as "the largest schism in a thousand years."
Following the sending of exarchs, the ROC officially "suspended relations" with Constantinople, a step that has been taken only once in recent history, when Constantinople granted autocephaly to the tiny, formerly Soviet nation of Estonia (with little over a million inhabitants in total) in the early '90s. At that time, however, the ROC and Constantinople fairly quickly came to terms and agreed to a solution that would allow two independent jurisdictions in the nation, one under the ROC, and one independent.
Following Constantinople's rehabilitation of Filaret and the UAOC, however, Moscow has in recent days taken the more drastic step of totally breaking communion and all clerical relations with Constantinople. This means not only that the Patriarch of Constantinople will no longer be prayed for in the Liturgy along with the other Orthodox Patriarchs, but also that all faithful of the ROC are forbidden to receive communion in Churches under the jurisdiction of Constantinople, and vice versa. In the past few days, officials of the Moscow Patriarchate have forbidden their faithful from receiving communion at any Churches ecclesiastically subject to Constantinople, even on Mount Athos, the world center of Orthodox monasticism--they have also publicly declared that the position of Primate in Orthodoxy, formerly held by Constantinople, simply no longer exists.
The question at this point is what happens next, on multiple levels. Constantinople has shown no signs of slowing their process, and in the near future they will likely finally and officially grant autocephaly to Ukraine; this will in turn likely trigger a response from Moscow, which may go so far as a formal act of excommunication or anathematization directed at the Constantinople Patriarchate.
On the ground, in Ukraine, the question is to what degree the granting of autocephaly actually unites the various Church groups involved. At minimum, the UAOC and the UOC of the Kyiv Patriarchate will be united with one another and with Constantinople. The ROC in Ukraine will also certainly continue to be out of communion with them, and will now no longer be in communion with Constantinople as well. The ROC in Ukraine, however, may, and almost certainly will, lose parishes and faithful to the new independent Ukrainian Orthodox Church. This process, though, has the strong potential to lead to judicial conflicts and even open violence, as faithful and clergy of different factions vie for individual parishes and all that goes with them. In the long run, this process may, or may not, lead to a decisive victory for one side or another; but in all likelihood both groups will continue to exist for the foreseeable future.
Outside of Ukraine, though, the situation is even more incalculable. The fact of the ROC being out of communion and actively hostile to Constantinople is in itself, by sheer force of numbers, significant; it means that the most sizeable group of Orthodox in the world no longer consider Constantinople to be the Primatial See of Orthodoxy and no longer participate in any of the common structures or projects spearheaded by Constantinople--which are, essentially, all of the common structures and projects that exist within Orthodoxy. Whatever Constantinople does apart from the ROC will be, by that fact alone, dramatically less significant, less prestigious, and less effective.
The real question, however, is how the other autocephalous Churches of world Orthodoxy choose to respond to this state of affairs, which presents them with an immediate and highly fraught puzzle. Within Ukraine itself, these Churches will be forced to decide which of the two jurisdictions to be in communion with; and outside of Ukraine, they will have to decide what, if any, steps to take in response to Moscow and Constantinople's breach in communion.
The most likely step for all these Churches, at least in the short term, is, paradoxically, simply to do nothing. It is far from unprecedented within Orthodoxy for a Church to be simultaneously in communion with multiple Churches that are not in communion with each other. There is, in fact, such a situation going on right now; the Patriarchates of Antioch and Jerusalem, the two most important Middle Eastern Orthodox Churches, have not been in communion with each other over a jurisdictional dispute for a number of years. No other autocephalous Church, so far as I know, has made any move in response to this; they have simply continued to simultaneously relate to and be in communion with both.
Of course, with Constantinople and Moscow, the issue is not nearly so simple. Jerusalem and Antioch are poor, marginal, and persecuted; both Constantinople and Moscow have strong ties of direct and indirect influence extending all across the world of modern-day Orthodoxy. It is practically impossible for any autocephalous Church to simply ignore either one, let alone both.
Leading up to this break, Moscow, at least, has been actively seeking support from other Patriarchates; by my count, the Patriarchs of Alexandria, Jerusalem, Serbia, and Bulgaria have all been brought to make strong statements of support for the ROC in Ukraine and against Constantinople's actions in recent weeks. The Orthodox Church in America, which has close ties to the ROC but is nominally autocephalous (an autocephaly granted by Moscow, and thus not recognized by Constantinople), has simply stated that they would not break communion with Constantinople. The Patriarch of Antioch has pointedly refused to endorse either side, but has bitterly complained about the divisions in the Church and the lack of attention shown to his own conflict with Jerusalem.
We are, really, in uncharted waters here. There are any number of theoretical ways this situation could resolve itself, but much depends on the precise actions and reactions of the parties involved. What is very close to certain is that neither side will, at this point, back down in their immediate conflict with one another: Constantinople will officially grant autocephaly to Ukraine, and Moscow will continue to try to cut itself loose from Constantinople.
Other autocephalous jurisdictions will try to mediate this conflict, but there is not really anyone in the Orthodox world with any significant amount of sway with either Patriarchate. A Council could be called, but it is extremely unlikely Moscow would ever attend such a meeting. Only a few years ago, Constantinople tried to summon a Pan-Orthodox Council in Crete, but was stymied by Moscow's unilateral refusal to attend (ostensibly for doctrinal and organizational issues)--a fact that ipso facto rendered the Council a dead letter. It does not seem likely Moscow would ever attend a Council where it actually had something to lose; and there is no longer an Orthodox Emperor to force them or anyone else to do so.
An Estonia-like solution is, of course, still very much possible; an agreement whereby both jurisdictions are allowed to exist in Ukraine simultaneously in communion with one another and with the rest of Orthodoxy. With every day that goes by, this becomes less and less likely, but it is still an obvious solution and can never be discounted. This resolution would presumably lead to at least a healing of the immediate Schism, though it would not bring an end to the deeper issues bringing Moscow and Constantinople into conflict. The status quo would be maintained, in other words, but not reconciled.
If the Schism continues and escalates to the point of open excommunication, whatever the other autocephalous Churches do, this would de facto lead to a situation where world Orthodoxy, over time, becomes more and more separated out into two clearly divided, hostile, competing institutions. Other autocephalous Churches could, in all probability would, be drawn into it and break communion with one side or the other. Or, if all of them maintain an ostensible neutrality, the labyrinthine, schizophrenic dimensions of modern Orthodoxy would be greatly increased. Either way, the ability for Orthodoxy to operate in any univocal or united way, or to ever join in on any common projects or statements, would be effectively neutralized.
In the long run, of course, regardless of what happens in the near future, one side or the other could "win out" in such a way as to render the other side impotent and of far less significance in Orthodox affairs. This does not seem particularly likely to me, but both sides have partisans who strongly believe in these scenarios, so they should not be entirely discounted.
The scenario by which Constantinople "wins" is fairly straightforward, and generally involves less "total war": due to on-the-ground conditions, the united Ukrainian Orthodox Church rapidly absorbs the ROC in Ukraine, and becomes too large and successful for any other Orthodox Church, including Moscow, to not to be in communion with. After the other autocephalous Churches gradually accept it as an equal and partner, Moscow is forced to accede, and the UOC thus takes its place in the pantheon of autocephalous Churches as a strong counterweight to Moscow's supremacy; and in the long run, a smaller, chastened, and more humble Moscow becomes a more reliable and less independent part of Orthodoxy under the authority of the Ecumenical Patriarchate.
The scenario whereby Moscow wins is also fairly straightforward, but rather more brutal: Constantinople has very few Orthodox believers under its direct jurisdiction, relying for its position and authority on soft power and prestige. If Moscow can successfully use its clout, which includes at least potentially the entire political and diplomatic power of the Russian state, abundant monetary funds, and numerous believers scattered throughout the world, to persuade other autocephalous Churches to join it in breaking from or even excommunicating Constantinople, at a certain point, Constantinople would simply become a tiny, poor, and insignificant Church, and a threat to no one. At this point, it would matter very little what they did or decided on any question; Moscow would be free to operate however they wished, whether, in the long run, they returned to communion with Constantinople or not. Moscow could even, depending on the situation, take the most drastic step of all: claim for themselves the Primacy ostensibly vacated by Constantinople.
As I said, I do not think either of these scenarios particularly likely, though they each have something to commend them.
Historical/Ecclesiastical Background
The difficulty with Orthodoxy as a modern phenomenon is always that it has never had a fixed Church order or ecclesiology. The Church of the Byzantine Empire had two institutional sources of unity at the highest, universal level, both of which modern Orthodoxy has no access to: the office of the Emperor, and the office of the Pope.
While it initially contained multiple Patriarchates with some degree of independence, following the Muslim conquests, Alexandria, Jerusalem, and Antioch were all lost to the Empire, while Rome and the Latin Church became gradually estranged from Byzantium first politically and later religiously; this led, in the long run, to an almost total ecclesiastical hegemony of Constantinople in the Empire.
Constantinople's status even as a Patriarchate, however, had been heavily disputed, to say the least. Prior to Constantine, there had been three Apostolic Sees generally recognized: Rome, Alexandria, and Antioch, all of which had personal ties to the Apostle Peter. Rome was, of course, the acknowledge Primate, as the See where Peter and Paul had shed their blood. Byzantium, on the other hand, was a minor city politically and ecclesiastically, with no even legendary ties to Peter or any other Apostles.
After Constantine re-founded Byzantium as his new capital, Constantinople or New Rome, the vastly increased political importance quickly translated into greater ecclesiastical importance as well. During the late 4th and 5th centuries, clerics of Alexandria and Antioch competed over and against Constantinople and to maintain their own primacy in the East, a conflict that led eventually to the estrangement of significant numbers of their followers from the Empire and the Imperial Church. Finally, at the Council of Calchedon, the assembled clergy, absent the Roman legates, passed a canon declaring that Constantinople, because it was the Imperial City, ought to be the second See in the Church after Rome. Pope Leo the Great, however, reacted angrily to this appeal to politics in Church order, and publicly annulled the canon, since Constantinople was neither an Apostolic See, nor could it possibly take precedence over the older Apostolic Churches of Alexandria and Antioch.
Still, in the long run, the close ties between Church and State in the Byzantine Empire, where the Emperor held ultimate sway in both spheres, made Constantinople's religious hegemony simply a fact of life, one that received little official legitimation for centuries, but which became more and more official over time and especially as Rome's role in the Byzantine Empire, for many centuries (particularly those of the so-called Byzantine Papacy) powerful and decisive, gradually waned.
When the Byzantine Empire fell, however, the believers and Church structures that had once been a part of it were brought into a state of disarray. Nations and ethnic groups were forced to fend for themselves, which they did in different ways. Some (the large proportion of Antiochenes who now form the Melchite Church, the majority of the Kyivan Rus who now make up the Ruthenian and Ukrainian Greek Catholic Churches, etc) re-entered Communion with Rome and became a part of the modern, post-Reformation Catholic world of multiple sui iuris Churches under the ultimate governance of the Pope; others remained in communion with Constantinople while learning to manage their own affairs, leading to the existence of "autocephalous" Orthodox Churches in the modern sense.
The Church of Moscow, however, adopted a rather more aggressive strategy, forging extremely close ties to the expansionist Muscovite Empire. Its "autocephaly" was gained essentially by force, including an incident where the Patriarch of Constantinople--who had traveled to Moscow to raise funds, since the Ottoman Empire maintained the Patriarchate only on the basis of compulsory extortion and simony--was held captive in Russia until he agreed to grant the Bishop of Moscow the title of "Patriarch." Along with this aggression in Church matters came the ideology of "Third Rome," which held that, just as Constantinople had been the First See in the Church not due to any Apostolic pedigree, but solely due to its role as the capital of the Orthodox Empire, so too, Moscow, the capitol of the Orthodox Russian Empire, had now inherited that same primacy. As Moscow and the Russian Empire expanded politically, so too did the jurisdiction of the Russian Orthodox Church, leading to many churches that had been formerly been subject directly to Constantinople now becoming subject instead to Moscow.
When the Russian Empire modernized under Peter the Great, so, too, was the Russian Orthodox Church modernized; in 1721, the Patriarchate of Moscow was actually abolished, replaced by a Synod subject to the Czar, and not reinstituted until 1917, in the throes of the Russian Revolution. The ROC, however, was soon, under the Soviets, officially abolished--and it was not until 1943, when Stalin made the decision, to benefit nationalistic sentiment against the Nazis, to revive and sanction Russian Orthodoxy, that the Patriarchate was officially restored once again. Under Stalin and his successors, however, the ROC was thoroughly penetrated and largely controlled by the KGB, even as it was marginalized and downplayed by the officially atheistic state. Since the fall of the Soviet Union and the rise of Vladmir Putin, however, the ROC has re-forged its close ties to the Russian State, receiving a great deal of support and approbation from Putin, and in return endorsing his policies and political affairs.
Constantinople, on the other hand, suffered after the fall of the Empire through centuries of repressive Ottoman rule--until the rise of the modern Turkish state following World War One, which paradoxically butchered its followers and drove them out of the country, reducing it to a tiny fraction of its former numbers, while at the same time by its official secularism giving it, for the first time in centuries, almost total freedom to operate independently on the global stage. This, for the most part, seems to be Patriarch Bartholomew's vision: of a restored Orthodox Church separate from, liberated from, the State and now operating in a unified way under the strong, independent leadership of Constantinople.
If this break is happening now, then, it is in large part because these two institutions are both in positions of relative strength--and also because their basic visions for the structure and order of Orthodoxy as a global institution are radically in conflict. Moscow, in a sense, is the less ambitious one; it wishes to ratify a Church order tied to political structures, and in particular to the ethnic-national Orthodox Empire embodied in the contemporary Imperialism of the Russian State, which sees all of its former holdings, whether nominally independent or not, as part of its sphere of influence and consequently under its spiritual and jurisdictional sway. Patriarch Bartholomew, on the other hand, ambitiously wishes to forge a modern Orthodox Church that exists as a universal institution independent of national, ethnic, and political divisions, ruled by a strong independent Primate whose status comes, not from current Imperial sway, but from past tradition, and divided in an orderly fashion into autocephalous Churches whose boundaries are dictated by those of nation-states.
The two groups also have more intellectual debates and alignments, the largest of which is how Orthodoxy ought to relate to other Christian groups--in particular the Catholic Church. Patriarch Bartholomew is famously ecumenical, and is personally close friends with Pope Francis, with whom he as cooperated on a vast number of projects. The Patriarch of Moscow, however, is generally opposed to rapproachment with other Christian groups; though he was finally brought to meet with the Pope for the first time in history in recent years, the ROC has always been the main foot-dragger and obstacle to cooperation between the Catholic Church and Orthodoxy.
Over Ukraine, the issue is also one of jurisdiction. Kyiv was formerly under the direct jurisdiction of Constantinople, which was eventually forced, by the political realities, to allow Moscow to ordain its bishops. Moscow thus claims Constantinople has no right to grant Ukraine autocephaly, since it is its own canonical territory. Constantinople, for its part, claims the exclusive right, as Primate, to grant autocephaly; it has, in the past, refused to recognize Moscow's attempts to grant its subject Churches this status. It also claims that Ukraine, regardless of the changes brought about by conquest or the grant in question, is properly its own canonical territory--and thus Moscow has absolutely no right to jurisdiction over it.
This is not an easy dispute to resolve, on any level. Constantinople's claim to a strong primacy independent of any Orthodox Empire is at least historically an innovation; Moscow's claim to absolute jurisdiction over the Church of another nation-state due to past Imperial rule is at base ruthlessly political: but the far deeper issue underlying both these stances and the conflict as a whole is that there is, in fact, no clear way to resolve any of these disputes, neither in terms of Church structure nor in terms of canonical jurisdiction, within the institutions and traditions of Orthodoxy as it has existed since the Fall of the Byzantine Empire.
The resolution of this dispute, then, depends in reality not on any intra-Orthodox ecclesiastical principle, but simply and solely on the will of the parties involved, as well as the chaotic conditions and fortunes of politics and history.
As a Catholic, all I can say is that we ought to pray, whatever Christian group we may belong to, for all those believers who are caught up in the middle of this fiasco: and trust, ultimately, in Christ, who loves his Church, and will unite all those who belong to him, finally and forever, in the Kingdom of his Father.