Hopefully someone with a better knowledge of early Christianity will correct me. At the beginning of Christendom, there were five church patriarchs: one in Constantinople, one in Rome, one in Antioch, one in Alexandria, and one in Jerusalem.

They were fairly autonomous figures, but the Bishop of Rome was considered the first among equals by the other four as the Bishop of Rome was the successor of Saint Peter.

Eventually there was the idea of papal primacy, where the Pope (who is the Bishop of Rome) was not just the first among equals, but the head of the entire church. "Catholic" roughly means universal church.

There were a lot of other contributing factors, but this was one of the contributions to the eventual East-West schism. Eastern Christianity more commonly goes by the name of the Orthodox Church, and is a lot more decentralized. There are many more patriarchs now than there were at the beginning of the schism, but the Ecumenical Patriarch of Constantinople--which is a cool title-- doesn't have analogous authority of the Pope but has the closest position in the Orthodox Church. He's still regarded as "first among equals" among the orthodoxy and you should listen, and consider what he has to say but other bishops can choose to do their own thing.

Though if the churches ever reconciled even the Ecumenical Patriarch of Constantinople has said the Pope would resume his status as first among equals.

So Rome became the seat, because in effect the Catholic church is the church that recognizes the Bishop of Rome as sitting in the seat of Power.

It's been a bit since I've talked about this, but I believe the majority of my post has been culled from what I read in Diarmaid MacCulloch's Christianity: The First Three Thousand Years.