In the early Church, as the age of the Apostles was winding down, and the bureaucracy of the Church was taking a more recognizable shape, there were basically three sees: Antioch, Alexandria, and Rome. I think many will argue that Jerusalem should also be counted, but Jerusalem did not exist, having been levelled by the Romans and rebuilt under another name. I think the local ecclesial community was centered more in Caesarea at the time, so it can be argued that the See of Jerusalem was vacant or in exile at the time, but in practical terms it didn’t really exist.
In time, as Christianity became more accepted, and especially after it was made the official religion of the empire, pilgrimmages to Jerusalem reestablished the prominence which that see was accorded.
Constantinople was obviously not a charter member of the club of Apostolic Sees, but since it was the capital of the Empire, and was duly given authority and recognition by Rome and the other Apostolic Sees, it is generally accepted that there are 5 historical Apostolic Sees.
Due to their territory being overrun by Muslims, and divisions between Catholic and Orthodox, the Sees of Alexandria, Antioch, and Jerusalem hold ceremonial prominence, but not the practical prominence they once did. Rome is obviously unrivalled in today’s world. Although the Orthodox Patriarch of Constantinople does not exercise immediate power over the other Orthodox churches, mostly in Eastern Europe, he is still seen as first among equals in Orthodox ranks. Moscow, due to the political importance of its territory, would like to be seen as the third great see in today’s world, alongside Rome and Constantinople. Other Orthodox, however, are not especially keen on this.