Great Schism

The Great Schism, also known as the Great Schism of 1054 or the East-West Schism was the schism by which the Roman Catholic Church and the Eastern Orthodox Church separated. Cultural and liturgical differences were debated as early as the 6th century; they officially split by excommunicating each other in 1054, and only ended the mutual anathema in 1965.

Official reasons for the separation
The official disagreements between the two churches were numerous and included seemingly trivial issues like baking and more critical issues of the nature of the Holy Spirit. For example, the Roman Catholic church insisted (without any clear biblical instructions that it should be so) that the Eucharist must be unleavened bread. The Eastern Orthodox Church accepted the use of leavened bread (however, the Jewish religious law forbids eating leavened bread during the Pesach, so Jesus and his disciples, being Jews, would have eaten unleavened bread during the Last Supper). There was also the more esoteric question of whether the Holy Spirit proceeds from the Father or from the Father and the Son - something known as the Filioque.

Actual reasons for the separation
Of course, like most things, "official positions" are rarely as important as power and politics. The two sides had many differences over this history of the church, with the appointment of John Chrysostom as Patriarch of Constantinople disputed by the Byzantine emperor as early as 404 CE, leading to the eastern church being excommunicated until 415 CE. There was another schism from 482 to 519, over who had ultimate doctrinal authority, with more excommunications.

Muslim conquests had reduced the Catholic territory, leaving only two (Constantinople and Rome) of the five sees (Alexandria, Antioch and Jerusalem had fallen). As the centre of gravity of Christendom shifted northwards and eastwards, there were disputes over who should have authority in the Balkans. Bishop of Rome, Pope Leo IX, thought he should be head of the remaining Catholic Body. The Patriarch of Constantinople, Michael Cerularius thought otherwise, believing that while the Bishop of Rome was the most esteemed, he still shouldn't be able to tell the other Patriarchs what to do.