Perhaps a muslim could answer better, but the issue stems from the fourth generation Caliphate Ali, who was assasinated by a rival political camp. After the assassination, some felt that the rightful successor should be a descendent of Muhammed, as the four previous "Rashudin" or rightly guided caliphs had been. This sect became known as Shias or Shiites. The other sect believed the leader should be a sort of elected official, that sect became the Sunni's.
Sunni's account for the bulk of muslims everywhere, but Iran is predominately shia and established a shia based theocracy under sharia Muslim law. Iraq has more shias than sunni's and their differences are one of many issues creating problems with the constitution.
Since the split both sects have also evolved and have new conflicts. One major conflict is the issue of jurisprudence, there are at least 4 or 5 different schools of jurisprudence out there and the interpretation of Muslim law is a major source of disagreement. Thats about all I can really say off top, Islam IMO is very difficult to understand because it is decentralized and also because for me I am Christian and learned very little about it until I was about 17.
Also theres a very good book by this Arab historian named Hourani, I believe its called a History of the Arab People's, at times it is dry and a little boring but is good especially with explanations on early Islam.