The catalysts of the first schism included:
Schism between Western and Eastern Churches
With movement of the emperor and political authority from Rome to Constantinople, a division was caused in the religious climate of the empire. Because of the custom of the emperor having more or less authority in both temporal and religious matters, this eventually caused a split between Bishops following the Pope in Rome and those following the emperor in Constantinople.
This lead to the exchange of excommunications by the representative of Pope Leo IX and the Patriarch of Constantinople Michael Cerularius in 1054 (finally rescinded in 1965) and the separation of the Roman Catholic and the Orthodox churches, each of which now claims to be "the One Holy Catholic and Apostolic Church." Though communion was not finally broken until after the Ottoman invasion of Constantinople in 1053 the fundamental breach has never been healed.
Schism inside the Roman Catholic Church
The second, and temporary, schism in the Western church resulted from the return of the Papacy from Avignon to Rome by Pope Gregory XI in 1378, ending the Avignon Papacy.
After Gregory XI died, the Romans rioted to ensure an Italian was elected; the cardinals, fearing the crowds, elected an Italian, Pope Urban VI in 1378; but in the same year the majority of them removed themselves to Fondi, and elected a rival Pope from there, Pope Clement VII. Later a council at Pisa was held in 1409 to try to solve the dispute, but it only resulted in the election of a third Pope, Pope Alexander V by the council, soon to be followed by Pope John XXIII.
Finally, the Council of Constance in 1417 deposed John XXIII and the Avignon Pope Benedict XIII, received the resignation of the Roman Pope Gregory XII, and elected Pope Martin V, thereby ending the schism.
From this time forward in the Catholic church it was decreed explicitly that no Council had power over the Popes, and there is no way to undo a Papal election by anyone but the pope.
The alternate papal claimants have become known in history as antipopes.
The Great Schism of the Western Church occupied the energies of John Gerson (q.v.), one of the great theologians of the age.