A quick note on the History of Islam
The history of Islam concerns the religion of Islam and its adherents, Muslims. "Muslim" is an Arabic word meaning "one who submits to God". Muslims and their religion have greatly impacted the political, economic, and military history of the Old World, especially the Middle East, where its roots lie. Though it is believed by non-Muslims to have originated in Mecca and Medina, Muslims believe that the religion of Islam has been present since the time of the prophet Adam. Muslims believe that prophets Noah, Abraham, Moses, Jesus, among others, were all Islamic prophets, and they have equal veneration in the Qur'an. The Islamic world expanded to include people of the Islamic civilization, inclusive of non-Muslims living in that civilization.
Ali, the cousin of Muhammad, and the first man to accept his "new" religion, is supposed by the Shia group to be the rightful person to succeed Muhammad. Shias provide various verses from Quran and Hadith(Sayings of the Prophet) to emphasize their rightfulness. Shia means follower, and Shia people claim to be followers of Ali.
However, the larger group, or the Sunnis - followers of Sunnah - named so for considering Sunnah (Prophet's way of life and his quotes) authorative as Quran, believe that Abu Bakr, father in love and one of the believers rightful to succeed Muhammad, as by consensus(Ijmaa).
All of the so-called "feud" started hence between Muslims. Later, Shias were sectioned into many groups, such as Zeydis, Ismaeelies or Sixers, Eighters, Twelvers or so. (This strange naming arises from the number of holy people known as Imams after Muhammad. Eighters believe in Eight Imams, ...)
Sunnis were fractionated into many schools, the major of it's surviving ones being: Hanafi (School of Abu Hanifeh), Maliki (School of Malik), Shafi'ei (School of Muhammad bin Idris, known as Shafi'ei) and Hanbali (School of Ahmad bin Hanbal).
Today, this issue is only discussed by scholars and intellectuals in Muslim world, and other groups of Muslims live peacefully by regarding their different vistae on the matter.