For almost all of period there was only one Christian Church, headed by the Pope in Rome. However, the Pope was viewed as first among a small group of bishops who, by virtue of where they had theeir see, were viewed as being slightly elevated among others. These were the patriarchs, and in period there were four apart from Rome (there are more now).
Locations of the Patriarchs
Constantinople was declared to be the second-senior bishopric of Christianity by the First Council of Constantinople in 381CE. The Popes in Rome, however (Damasus and Gregory the Great), refused to confirm this canon. With the schism in 1054 between the Roman and Byzantine Churches, two lines of patriarches, the Latin and the Orthodox were established. In 1204 when the Crusaders seized Constantinople they imposed their own patriarch, of the Latin Rite, and in 1215, by the Fourth Lateran Council, the Roman church accepted that its patriarch was second only to Rome. The Latin line was dispossessed in 1261; however in 1439 the Council of Florence (whilst not recognised by the Orthodox Church) recognised the Orthodox patriarch as second to Rome.
The patriarchy of Alexandria is claimed to have been established by St.Andrew in 42CE and its holder bears the title of pope alongside the Roman patriarch. From foundation until 451CE there was a consistent line of patriarchs; after that date there were two lines, one of the Orthodox Church, the other of the Coptic Church. It does not appear that the Crusaders ever attempted to impose a Latin patriarch in the city.
From circa50CE until 544 there was a single line of patriarchs in Antioch; after that date there are 2 parallel lineages, one for the Eastern (Greek) Orthodox Church, the other for the Syrian Orthodox Church. From 1100 to 1268, during the Crusading period, there was also a Latin Patriarch in the city, established by Bohemund, first Prince of Antioch. During this period the Greek Orthodox patriarch was forced into exile in Constantinople. In 1342 Patriarch Ignatius II transferred the seat of his office to Damascus, where it has since remained.
The first patriarch of Jerusalem was James the Just, brother to Jesus Christ. In 1054, with the Schism between the Roman and Orthodox Churches, the patriarch adhered to the Greek side, but in 1099, with the capture of Jerusalem by the Crusaders a Latin patriarch was installed. In 1187, with the loss of Jerusalem itself, the patriarch moved to Acre, and in 1291 to Cyprus, when Acre fell. Finally in 1374 Rome appointed titular patriarchs, who sat in Rome. Orthodox patriarchs continued in Jerusalem thoughout period.