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This site has a good overview of the history from a Catholic viewpoint.
Here is a high level overview based on that site and the article on Wikipedia (which currently stands in question of its neutrality).
After the collapse of the Western Roman Empire, the church headquartered at Constantinople began to have ongoing squabbles with Rome due in some part to the Emperor's meddling in Church affairs.
In 1054, after the Norman conquest of southern Italy, the friction escalated due to the imposition on the Greek colonies there of the practice of Eucharist using unleavened bread.
A Papal legate was sent to address the situation, but (apparently largely) personality issues caused the two delegates to mutually anathemize each other (but not their respective churches). This was not considered very momentous at the time, but in retrospect it marked the departure point between the two bodies.
Even after that, there were future attempts at reconciliation. The collapse of the Byzantine Empire, between 1450 and 1490, precipitated the enduring split.
With the last few popes, there have been renewed attempts at reunification. In fact there are a number of Eastern-Rite churches in communion with Rome (e.g. the Byzantine Catholic rite). But it seems that the laity of the Eastern Orthodox are more opposed to reunification than the leaders, at least in some ways.
An important aspect of this the differing views, from an early age, of the role of the Pope. Easterns regard him as "first among equals" or to have primacy, while Westerns, of course regard his role as one of authority.
The Filioque question was an early (sixth century) bone of contention between the East and the West. These days, Rome considers the two to be equivalent statements and it seems likely that it would be a small or nonexistent issue if serious reconciliation were ever to take place.