The detail
Long before 1054, the Greek East and Latin West had been drifting apart. They spoke different languages, developed different theological styles and customs, and from 800, when Charlemagne was crowned emperor in Rome, lived under rival claims to the Roman Empire. Rome and Constantinople had already been out of communion several times, for example during the Acacian schism (484–519) and the Photian dispute (863–867). The main disputed issues were the authority of the pope over the whole Church, and the Filioque, the phrase "and the Son" that Latin churches added to the Nicene-Constantinopolitan Creed (usually traced to the Third Council of Toledo, 589; promoted by the Frankish court; sung in the Creed at Rome only from 1014). Practical differences also mattered: the West used unleavened bread (azymes) in the Eucharist, required celibacy of priests, and fasted on Saturdays.
The crisis of 1054 was triggered by quarrels over Latin and Greek customs in southern Italy and Constantinople, including a letter by Leo of Ohrid criticising azymes and the closure of Latin churches in Constantinople by Patriarch Michael Cerularius. Pope Leo IX sent legates led by Cardinal Humbert of Silva Candida. Negotiations broke down, and on 16 July 1054 Humbert laid a bull of excommunication against Cerularius and his supporters on the altar of Hagia Sophia. A few days later a synod under Cerularius anathematised the legates. Leo IX had died in April, so the legates' authority was itself questionable, and the sentences named individuals, not the two churches as such. At the time, many did not regard the event as decisive.
What made the break real were later events. The Crusades brought Latins and Greeks into close and often hostile contact; Latin residents of Constantinople were massacred in 1182, and in 1204 the Fourth Crusade sacked Constantinople, installed a Latin emperor and a Latin patriarch, and placed Latin bishops in Greek sees. For many Orthodox, this, more than 1054, sealed the separation. Two reunion councils followed. At the Second Council of Lyon (1274) Emperor Michael VIII Palaiologos accepted union largely for political reasons, but it was rejected by clergy and people and abandoned after his death (1282). At the Council of Ferrara-Florence the decree of union Laetentur caeli (1439) was signed by most of the Greek delegation, with Mark of Ephesus refusing; it was rejected in the East, proclaimed in Constantinople only in December 1452, months before the city fell to the Ottomans (1453), and formally repudiated by an Orthodox synod in 1484.
Theologically the two sides tell the story differently. In the Orthodox account, Rome departed from the common faith of the first millennium by altering the Creed without an ecumenical council and by claiming a universal jurisdiction never recognised in the early Church, where Rome had a primacy of honour among the patriarchates; the Orthodox Church sees itself as the continuation of the undivided Church. In the Catholic account, communion with the successor of Peter is a constitutive mark of the Church, the Filioque expresses legitimate Western teaching compatible with the Greek "through the Son", and the Eastern churches separated themselves from that communion, though they kept valid sacraments and apostolic succession. Both narratives agree that politics, mutual ignorance and violence did as much as doctrine to deepen the rift.
In the 20th century relations improved markedly. Pope Paul VI and Ecumenical Patriarch Athenagoras I met in Jerusalem in 1964, and on 7 December 1965 they issued a joint declaration "erasing from memory" the excommunications of 1054. This lifted the personal sentences but did not restore communion. A Joint International Commission for theological dialogue began in 1979–1980; its Ravenna document (2007) addressed primacy and conciliarity. Pope John Paul II expressed regret for the 1204 sack in Athens (2001). The main open questions remain papal primacy and infallibility, the Filioque, and the status of Eastern Catholic churches (a sore point since the Union of Brest, 1596).