I will be the first to admit that Islam and its theology is a major gap in my knowledge. I believe the following to be accurate, although I would be quick to be corrected if it is not.
January 29, 661. Kufa, Iraq. A man praying in a mosque is assassinated by a religious radical. This event sounds like it could be a modern headline, but the dead man is the Caliph Ali ibn Abi Talib - the son-in-law of the Prophet Muhammad. Few know it at the time, but the murder of Ali sows seeds that will one day divide the Islamic world in two - the Sunni and the Shi'a.
After the death of Muhammad in 632, the question of who would be the next Caliph (a close approximation would be "steward" or "successor") had been a heavily disputed issue. It was generally accepted that Muhammad's successor should be a close follower and a member of the Meccan Quraysh clan; multiple men, however, vied for this role. The most prominent were Muhammad's companions Abu Bakr, one of his first converts; Umar, the famed legal scholar and intellectual; and Ali, the husband of Muhammad's daughter Fatima and the first man to convert to Islam.
Abu Bakr and Umar, with their financial and patriarchal support, were able to seize the reins of the Caliphate; Ali initially resisted but was forced to give in. Ali had many partisans, and many Muslims felt that Muhammad had intended for him to succeed to the leadership of the ummah (or the whole body of Islamic followers). However, Abu Bakr and Umar had much higher standing with the Arab community; Abu Bakr was the first Caliph to succeed Muhammad, and Umar the second. Their reign, and that of their successors, would be called the Rashidun Caliphate.
Under Abu Bakr; the Muslims unified the Arabian Peninsula. Under Umar, they struck out into the lands of the Byzantine and Persian Empires; by the time of his death in 644, the Arab conquest had swallowed Egypt, Syria, Israel, Iraq and Persia in a series of lightning wars. The Arabs, formerly a fringe tribe on the borders of the great empires that ruled from Constantinople and Persia, were now the great power in the Middle East.
The Rashidun Caliphate remained unsettled, though. Ali, resentful over being displaced, still desired to take the reins of his father-in-law's religion and empire. When Umar was assassinated in 644, a council of six ministers was appointed to chose a successor. Among them was Ali, but also Uthman, yet another son-in-law of Muhammad. When the council voted, Uthman voted for Ali - but all the others, fearing Ali's ambition, voted for Uthman. Ali had been denied again.
Uthman's reign was marked by rebellion against what some saw as greedy and craven ways. Ali, of course, was the favorite candidate among these rebels. These conflicts between the Arabs were driven as much by familial and property conflict as by religious or political principle; despite the ambitions of the Islamic code and tenets, the tribal divisions among the Arabs were hard to overcome.
In 656, Uthman died, and Ali inevitably rose to become the fourth Rashidun Caliph. Ali had been in conflict with Uthman and the previous Caliphs for what he saw as violations of Islamic law and abuse of power, but their supporters were powerful as well. Among Uthman's supporters was Muawiyah, a powerful nobleman and early opponent of Islam; Muawiyah rebelled against Ali in 657. This military struggle - the First Fitnah - was punctuated by military campaigns, arbitration, and attempts at peace before matters came to a head.
In 661, Ali was praying at the Mosque of Kufa, 100 miles south of Baghdad. A member of the dissident Kharijite faction slipped into the mosque and with a poisoned sword struck the Caliph down as he knelt. Even as Ali was dying, he told his sons that the assassin should be forgiven if he did not die; if he did perish, they should only pay like unto like and not persecute the man's family. When Ali died on January 29, 661, his son paid like unto like.
With Ali's death, Muawiyah became Caliph by default, beginning the Umayyad Caliphate - among the most famous and powerful of the Islamic religious monarchies. With its capital at Damascus, it produced some of the greatest art, philosophy, and poetry in the history of the world; its religious scholars set the course for the future of most of Islam. They referred to the "sunnah", the set of traditions and practices followed by Abu Bakr and Umar in accordance with Arabic law and inheritance. They believed that Muhammad had left no clear successor, and that it was up to his followers to decide who should lead them.
Since they followed the "sunnah", it was only a matter of time before they called themselves Sunni.
The followers of Ali, however, did not vanish. Many still believed that only a descendant of the Prophet Muhammad's family could ever be a true Caliph, and their ideas were not extinct. They believed that Abu Bakr, Umar and Uthman were all usurpers; Ali had been the only true Caliph after the death of Muhammad. They continued to revere Ali's sons and descendants, and would become called the "shi'atu ali" - the "party of Ali".
This would become shortened, in time, to the catch-all term "Shi'a."
To this day, those who believe in the descendants of Muhammad as rightful leaders of the faith, those who revere the line of Ali and its many successes and failures throughout Islamic history, have been known as Shia Muslims.
A good work on the history of Islam is Karen Armstrong's Islam: A Short History (New York: Modern Library, 2000).