Schisms in General
A "schism" is a rending, a tearing away. "Formal" or "open" schisms typically have roots which go back further than their obvious manifestation. I think it can fairly be said that there are always new "opportunities for schism" or "new seeds" being planted for future schism. Fortunately, by God's grace, these often come to nothing, and are more or less resolved.
One historical example of this was the "Easter controversy" in the second century, where the date for the celebration of Pascha (which differed between certain Eastern Churches and Western ones - the former typically using the Jewish Passover as their date, while the Romans and other westerners always kept it on a Sunday) became so contentious that the Pope of Rome (St.Victor) was threatening to excommunicate those who did not come over to the Roman usage. The problem was further frustrated by the fact that both sides claimed an Apostolic origin for their respective practices. Fortunately, cooler heads prevailed, and for the time being everyone "agreed to disagree", until the matter was eventually settled by the first Ecumenical Council.
Right now, we are (imho) living in dangerous times, precisely because there are plenty of significant opportunities for major schisms to erupt if certain people continue to act outside of a genuinely catholic frame of reference, and pursue their own agendas without adequate consultation of other local Churches. Hopefully, it will all come to naught.
Earliest Period of the Roman Church
The Latin schism had very deep roots, some of them going back (imho) to the decades following the "legalization" of Christianity itself under St.Constantine. If one cares to read the pre-Nicean Fathers, something becomes quite conspicuous - there is generally an absence of "Papal claims" as such. While the Roman Church is spoken of with honour, even this is framed differently than it will be in later centuries; it is so honoured because of it being a major Christian centre consecrated by the blood of so many martyrs. Whenever it's "apostolic origins" are cited, they unfailingly mention (and emphasize) both Sts. Peter and Paul.
The closest thing we have to a "pro-papalist" argument in the ancient Fathers, is in one place in the writings of St.Irenaeus's Against Heresies, where he is often quoted as saying that the faith of all of the Church's throughout the world is somehow measured against Rome (the idea drawn from this being that Rome is somehow "the standard", that communion with her is necessary to be a genuine Christian.)
Of course, the problem with this is two fold. On one hand, all that remains extant of Against Heresies is the Latin translation - the original being written in Greek. More significantly though, is that recognizing this, and examining the awkward Latin of the extant translation, there is plenty of reason to believe St.Irenaeus' actual meaning has been misunderstood. The context of the passage itself indicates as much, if you consider the "set up" in the text for the remarks about Rome (namely, that it would take too much time for the author to have given a listing of the apostolic succession of all of the various local Churches throughout the world, showing them to thus be genuine as opposed to the self consecrating gnostic heretics he was fighting). Many now believe, his point was that Rome reflected the Catholic (Universal), Orthodox faith of the Church - that she received the testimony not only of herself, but of all of the other Orthodox Churches throughout the world (which was quite hard to avoid, given that she was the Imperial capital, indeed pretty much the centre of the civilized world at that time), particularly in the form of the martyrs shipped from throughout the world, to Rome, to be put to death.
Thus the one so called "clear" advocacy amongst the early Fathers for those later Latin claims, is hardly so clear at all. Other oft cited (though "less unambiguous") texts tend to be cited as well, such as St.Clement's Epistles to the Corinthians. The problem with these texts of course, is that besides not advocating anything resembling the "papism" of later centuries (and certainly not the craziness of the post-schism Latins, including so called "infallibility"), there is actually some good reason to believe that when St.Clement was writing them, he was not even a Bishop, or at least not the ruling Bishop of the Roman Church (and hence "Pope", though such a title did not formally exist at that time.) One reason for believing this, is that most of the traditional "listing" for Bishops of Rome, place St.Clement's reign of the Roman Church toward the last years of the first century (his reign did not last long; he died a holy, martyr's death). Secondly, there are two elements in St.Clement's writings which indicate he was writing more towards the close of the 60's A.D., than towards the end of the first century itself.
(a) He speaks of the temple sacrifices (in Jerusalem) as something contemporary to him; well, the temple of Jerusalem was destroyed in 70 A.D.
(b) He speaks of the martyrdoms of Sts.Peter and Paul as having been relatively recent events, and also speaks of the circumstances of their martyrdoms (indicating that they were somehow betrayed, perhaps by someone within the Roman Church itself). Well, most traditional hagiographies place the joint martrydoms of these two Apostles in the 60's A.D., the decade before the temple of Jerusalem was destroyed.
Given this, it would seem pretty clear (to me at least, as well as many others) that St.Clement was writing in the mid-late 60's A.D., just before the Romans leveled the temple in Jerusalem.
Unfortunately, any involvement by the Roman Church in the affairs of it's neighbouring Churches (or beyond) is viewed by papism-apologists as being some kind of endorsement of later developments. However this makes no sense, since Church history is filled with many examples of one Church involving itself (rightly or wrongly, depending on the situation) in the affairs of another local Church. It seems though, when a Roman Bishop (or member of the Roman Church) does this, a lot more is read into it.
By "middle period", I mean the centuries after the Edict of Milan (313 A.D., when St.Constantine legalized the Church and restored Her property to Her, as well as endowing Her with many more), ending with the period of time others here have already spoken of (when the new Germanic lords of western Europe, ruling amongst the ruins of what once was the western part of the Roman Empire; some here have cited Pepin, which seems correct to me, though most cite the rise of "Charles the Great", aka. Charlemagne, and his coronation as "Holy Roman Emperor" by Pope Leo III as more decisive.)
During this "middle period", I think you see a lot of jockeying for power and "ecclessiastical rights" by many parties. Unfortunately, such avarice and defending of one's perceived "turf" has never really come to an end, and there are many relatively recent events in the history of the Orthodox Church indicating as much (indeed, the whole situation of western "juristictionalism" is a clear example of this.) The Popes of Rome were certainly not unique in this regard; many were indulging these ambitions in some fashion, with different ways of arguing in their favour.
It was also this period which saw the rise of a new perspective on the Church Herself. Where as previously, St.Ignatius of Antioch's statements on ecclessiology were pretty much the norm (that each local Church, overseen by a Bishop, celebrating the Holy Eucharist, embodied the "fullness" of the Church), in this new situation there was an increasing emphasis upon the Church as a universal body - seen from an outward, "above" perspective... a "collection" of local Churches, all cumulatively comprising a single, larger Church. St.Cyprian of Carthage was probably the one key Father to grapple with this subject; interestingly some Papal-apologists cite his works as evidence of their ecclessiology, though this should be perceived as odd for obvious reasons (namely, the fact that St.Cyprian himself thumbed his nose at a Pope in what some might call an early "test case" for Papal claims.)
This also seems to be the period where gradually (and it was, I think, precisely because of the above juristictional disputes) you see the Romans (who were slowly becoming "Latins" as well - since in the earliest period they were Greek speakers) begin to drop mention of St.Paul, when it claim to bolstering the claims of their local Church. Also, slowly something else also changed - it was no longer their "Church" (the Roman Church) which was so honoured and held in reverence, but the Bishop of Rome himself. This of course mirrored the juristictional jockeying of other hierarchs in Christendom at that time, so it was not a uniquely Roman failing.
This period also seems to be the one where we see the beginings of very exalted arch-episcopal titles, which if taken too seriously, imply things which are actually contrary to the Orthodox faith (for example, St.Gregory the Great's concerns over the title "Ecumenical Patriarch" being bestowed upon the Patriarch of Constantinople).
It was also toward the end of this time, that some of the more provincial (and divisive) changes occured in Roman liturgical custom. At the earliest, it may have been toward the end of this "middle period" that the epiklesis was dropped from the anaphora/canon of the Roman Liturgy. It was also likely in this "middle period" that the Roman Liturgy gained unique traits not found elsewhere, like it changing in some substantial ways depending on what "liturgical season" it was being celebrated in (this is a trait which would eventually spread throughout all of the various western liturgies, before they were finally suppressed and replaced with the Roman Rite or it's various local recensions.) Not all of these changes were objectively bad, but they did not necessarily "help" things either (because of the unfortunate human tendency to want to associate with people superficially like ourselves.)
It's in this period (IMHO) that the seeds of eventual, long term schism are to be found.
The Latter Period
I won't say much about this period, as many both on this forum and elsewhere have written about it. "Unique" changes in praxis/discipline continued in Rome, which was now in a cautious/fearful relationship with it's new (Germanic) secular lords. Things like the so called "Donation of Constantine" make their appearance during this time - a fraud to be sure, but one which oddly enough likely originally intended for use against the Franks and not Orthodox Christians to the East (basically, scholars like the late Fr.John Romanides argued that the so called "donation" was an attempt to keep Frankish hands off of the Papal domain, and to ensure the Popes had some measure of independence from the Germanic Emperors.) Sadly, that it was a fraud was forgotten, and eventually it just became another piece of poorly argued evidence in favour of Papism.
This was also a period of growing corruption in Roman Church - in the final century of the first millenia, some profoundly obscene situations had developed in regard to the Papacy and it's occupents. Eventually the political ambitions of the German Emperors conquered and occupied the Roman See itself in the form of "German Popes", and the antagonism of the Franks toward the actual Roman Emperors of Constantinople (and the "Eastern" Orthodox Church, which was rightly perceived to be a friend and booster of this Emperor) now won out. Thus it's not a coincidence that soon the "filioque" would be officially accepted by the Popes, and a lasting schism would begin (first with Pope Benedict VIII not being entered into the Diptychs of Constantinople - later, and more significantly, by the pseudo-excommunication by Cardinal Humbert of the Ecumenical Patriarch.)
After this "great schism", Rome would continue on it's "merry" way, adding error upon error. "Purgatory" became a dogma amongst the Latins, understood in a sense which the Orthodox world could not accept. Papism only grew, culminating in the 19th century with the First Vatican Council (where the Popes became "infallible" - oddly enough, they needed a Council of Bishops to tell them this and declare it a matter of fact...wrap your mind around that one). Liturgical deviance grew with the Latins, and in some respects in ways which are unnacceptable to Orthodoxy. "Indulgences" entered the picture, and strange over simplifications of the economy of salvation (which in a way, the "indulgences" heresy depends upon.) Latin minimalism/legalism took hold, and you saw the begining of things like "baptism by pouring", etc.
Of course, the West did not know even a "worldly peace" during this time of ever growing falsehood. The various Protestant revolts broke out, undoubtedly spurred on by circumstances the Latins themselves had created, though sadly these revolts did not improve the situation, but typically "threw the baby out with the bathwater", and introduced monstrous heresies into the world of their own. All of this gave rise to secularism and popular atheism; basically people of some intelligence got sick of the a-historical madness which western Christendom had often degenerated into, and so committed themselves to the ultimate "Protestantism", and simply abandoned God entirely, confusing the fantasies (idols) of heresies with the Living God.