Well, first of all it seems like your asking about 3 separate (though obviously related) large topic areas. I'm not sure what you think you mean by Early Christianity - do you mean 1st century, or 1st and 2nd, or 1st through 5th? People's definition varies.

I'll briefly run through the First Council of Nicaea and its surrounds, and then the Great Schism.

Nicaea 325, its causes, occasion, and aftermath.

To understand Nicaea, there's more than a few things to get into our heads. Let's begin with the catalyst for the theological controversy that engulfed much of the church (in the Empire) for most of the 4th century. Around 318 a presbyter in Alexandria (Egypt) objected to the local bishop, Alexander, about his preaching. Alexander was proclaiming a doctrine of eternal Generation - that the second person of the trinity existed eternally as Son and was co-eternal with God the Father. To Arius, this appeared heretical - the equivalent of saying there were two principles of ultimate being in the world, and thus 'two gods'. Arius taught, as best we can gather primarily from the writings of others, that the second Person (= of the Trinity), came into existence at a point 'later' (in time or 'before' time, but still 'later' than eternity-past, and so was created, and subordinate.

This led to controversy in Alexandria, the excommunication of Arius, and to Arius appealing to other local and nearby clergy for support of his position as the historic belief of Christianity. Alexander had little choice but to do the same - call for support from bishops in other parts of the empire to uphold his understanding of how the Trinity worked.

Now, keep in mind that precisely at this time Constantine was taking control of the Empire. And declaring that he was a Christian. The combination of those two factors, and the fact that Constantine had already been called on to intercede in the Donatist affair, meant that this 'local' problem quickly became a rather large issue and Constantine was resolved to bring it to a solution. Hence the Council of Nicaea. Actually, it was originally slated to occur at Ancyra, but was moved to Nicaea. Reports indicate somewhere between 200 and 318 bishops attended, almost entirely from the Greek-speaking East. The main, but not the only issue, on the agenda was 'dealing with this Arius matter.' Which is what it did - it formulated the Nicene Creed, which contained explicit language designed to exclude Arius' theology, and it was followed by several statements of anathema - things heretical and to be condemned - which individually targeted and excluded certain key statements of Arius.

For all intents and purposes, that was the effective end of Arius and his theology. But it also acted as a catalyst. It made clear that within the broader church there were some significant disagreements about how the Son relates to the Father within the Trinity. Those divisions were sparked into flame by Nicaea and occupied a great deal of theological debate and politics over the rest of the fourth century, until they were effectively resolved by (a) the accession of Theodosius I in 379 and (b) the council of Constantinople in 381

The Great Schism

The tendency is to think of the Great Schism (GS) as just something that happened in 1054. But really, the divisions between the eastern and western churches within the Roman Empire go much further back. In fact, even in the above period, the 4th century, those divisions are beginning to form.

Perhaps the single most influential non-theological factor is simply that the Western church functioned and developed primarily in Latin. While western theologians often were influenced by Greek-language theology, the reverse was far less true. The eastern half of the empire was far more populated, more developed, and 'Christian', and Greek-language theologians rarely received 'influence' from Latin works.

Furthermore, in the West you had the development of the papacy. The growing centrality and importance of the bishop of Rome in the 4th to 6th centuries strongly shaped Xianity in those region. The eastern churches never accepted that the bishop of Rome had anything more than a place of honour, they certainly did not accept the developing claims to be head of the church and seat of authority and derivative power in the place of Peter. This put east and west in tension.

Another factors is that the western empire ceased to exist thanks to the Germanic Migrations. This fundamentally altered the social structures in Italia, Gallia, Hispania, and North Africa. It moves western Xianity into the early middle ages. The eastern church didn't experience this same kind of disruption - in some ways the Byzantine church never had a 'middle ages', having much more continuity and stability as the successor empire to the Roman Empire. (Indeed, the Byzantine empire was the Roman empire. While the rise of Islam was disruptive, this was an external factor, not an internal situation.

Then you have the political complications of Byzantine territory in Italy, competing claims to jurisdiction, the now long-standing and supreme claim of Rome to monarchic authority, against the Byzantine view of Christendom as one emperor, one empire, one church. All of which came to conflict in the 11th century with two forceful personalities, Leo IX of Rome, Michael Cerularius of Constantinople. Michael attacked the Latin church as practicing heretical things, Leo responded by asserting Rome's authority and stating that Constantinople was Rome's daughter church. Eventually Leo excommunicated Michael, which obviously didn't go well.

That's the short version. Just like Nicaea, there's a long version. And a longer version. History is very often about 'level of detail'.

Feel free to ask follow up questions, very happy to answer or guide you to other materials.