From a Doctrine standpoint, what kind of difference is there between the various branches of Islam?
From what I remember, after the death of Muhammad (632 AD/CE), the caliph (successor, a position with both spiritual and temporal power) was Abu Bakr, Muhammad's father-in-law, and then was succeeded by Omar and Uthman. Many Muslims believed Muhammad's son-in-law Ali ibn Abu Talib should have succeeded. The supporters of Ali were the Shia. Ali took power after the death of Uthman, but a rebellion led by Muhammad's youngest widow Aisha (Abu Bakr's daughter) and by a Muslim general named Muawiyah. Muawiyah murdered Ali at a peace treaty and became caliph, founding the Umayyad dynasty.
From then on, Muslims divided between the side that favored the caliphs (Sunnis) and the side that favored putting Muhammad's descendants via Ali on the throne (Shia).