Even now, many people who hear these terms daily on the news are confused about what the real differences are between Sunni and Shia Muslims, so I, having been brought up in a very devout Shia household in Pakistan, thought I would explain these things, at least in rough terms. Here goes:
It all started hours after Mohammad's death: while his son-in-law (and first cousin) Ali was attending to Mohammad's burial, others were holding a little election to see who should succeed Mohammad as the chief of what was by now an Islamic state. (Remember that by the end of his life, Mohammad was not only a religious leader, but the head-of-state of a significant polity.) The person soon elected to the position of caliph, or head-of-state, was an old companion of the prophet's named Abu Bakr. This was a controversial choice, as many felt that Mohammad had clearly indicated Ali as his successor, and after Abu Bakr took power, these people had no choice but to say that while he may have become the temporal leader of the young Islamic state, they did not recognize him as their divinely guided religious leader. Instead, Ali remained their spiritual leader, and these were the ones who would eventually come to be known as the Shia. The ones who elected Abu Bakr would come to be known as Sunni.
This is the Shia/Sunni split which endures to this day, based on this early disagreement. Below I will say a little more about the Shia.
So early on in Islam, there was a split between political power and religious leadership, and to make a long story admittedly far too short, this soon came to a head within a generation when the grandson of one of the greatest of Mohammad's enemies (Abu Sufian) from his early days in Mecca, Yazid, took power in the still nascent Islamic government. Yazid was really something like a cross between Nero and Hitler and Stalin; just bad, bad in every way: a decadent, repressive dictator (and one who flouted all Islamic injunctions), for whom it became very important to obtain the public allegiance of Husain, the pious and respected son of Ali (and so, grandson of Mohammad). And this Husain refused, on principle.