In answering this, we must first note that all of our accounts of the actual splits between the Sunni and the Shia come from Sunni sources. This may mean that we do not have the whole story of what really happened. In general, though, we believe that the split between these two parts of Islam dates back to the death of the Prophet Muhammad.
After Muhammad died in 632, there was controversy over who should succeed him. There were two main schools of thought among Muslims at the time. One school held that Muhammad should be succeeded by his closest male relative. The other held that Muhammad’s successor should be chosen on the basis of a consensus among the leading males in the Muslim community. The closest male relative of Muhammad was Ali, who was married to Muhammad’s daughter Fatima. Ali was also Muhammad’s cousin. On this basis, he claimed to be Muhammad’s successor. However, his claim did not prevail. Instead, the leaders of the community decided that Abu Bakr, the father of Muhammad’s favorite wife, should be the next leader.
Ali’s supporters were very unhappy with this. Ali did eventually become the caliph, but he was assassinated and the caliphate was not passed down to his sons. The Shiites are those who believe that Ali was Muhammad’s true successor. The Sunni believe that Abu Bakr was the true successor. Since that time, other factors have arisen to drive the two sides apart, but this was the origin of the schism.