First of all, @LangLangC's answer is excellent. I intend only to expand on it.
The unified Eastern (Greek-speaking) and Western (Latin-speak) churches called themselves "orthodox" as opposed to the many heresies (like Arianism). They also called themselves "catholic" -- universal. They were united in religion, but administratively united only in theory, as they were much more fragmented administratively than we, today, think a unified church would be. Travel was hard even in the still-thriving Eastern Empire, and in the West, travel was very slow.
I expect that the old language of being "in communion" probably better describes the situation. Each city had its bishop and one or more churches. So you'd have the church of Milan, the church of Hippo, the church of Antioch, etc. The churches which considered themselves orthodox were "in communion", that is they recognized each others' sacraments, shared scriptures and confessions, recognized each others ordinations and accepted each others' members when travelling. They generally adhered to one or another of the ancient patriarchates.
So the church then was a much more decentralized thing than the church today. Note that this was mainly the effect of difficult travel -- especially in the Byzantine Empire the Emperor and the Patriarch of Constantinople exercised as much control as politics would let them. In the West, Rome's authority was necessarily entirely moral, and the East tended to forget that Rome still existed.
Some of the split between East and West is nearly as ancient as the Church itself. There was always a split in language and Constantine, the same Emperor who legalized the church, also was the Emperor who split the Empire!
Now fast forward six hundred years as the East climbs to the peak of Byzantium, while the West sinks to the worst part of the Dark Ages -- and then it all reverses, with Islam conquering one piece after another of the Eastern Empire, and the new, vibrant Medieval civilization is growing in Western Europe. The East resents the barbarian West trying to tell it what to do. The West resents the effete, snobbish East trying to conquer it and tell it what to do. And communication between the two got even worse as the Mediterranean got even more infested with pirates and travel grew even more difficult.
But everyone agreed that they were one church -- as long as you don't try to tell us what do, but do follow our instructions…
The final schism in 1054 was simply the last step of administrative separation when the hierarchies of the Eastern and Western churches excommunicated each other. (They were no longer "in communion".) This was basically an administrative thing. They continued to recognize each others' sacraments, ordinations, etc., but declared it to be unlawful to participate in each others' services.
As it happened, the Eastern church kept the title "orthodox" and the Western church kept the title "catholic" -- as far as I know, that was essentially historical accident, as the modern Roman Catholic and Eastern Orthodox churches both describe themselves as "one, holy, catholic and apostolic" and as "orthodox." (It's possible that the East called itself Orthodox because the richer East was the source of most of the heresies and orthodoxy was more of an issue there.)