The modern spelling of Shiite is Shi'i. Literally "the followers" or "the partisans". Short for shi'at Ali or "the partisans of Ali". The Shi'i are the subgrouping of Muslims who believe that the only proper leaders of the faith, or Imams, are direct blood-line descendants of Muhammad, through his cousin and son-in-law Ali ibn Abu Talib. The Shi'i are historically a minority in the Islamic world, constituting perhaps 15 percent of all Muslims, and rarely coming to political power. There are only a few notable Shi'i states, most prominently medieval Fatimid Egypt and modern Revolutionary Iran.
While the popular Western image of the Shi'i are of unpleasant, violent, ultra-orthodox fanatics with poor senses of humor, the Shi'i are actually considered on the verge of heresy by many mainline sunni Muslims, and whatever reputation for violence they have comes from the Iranian state, rather than the faith itself. While they are not technically heretics because they observe the Five Pillars of Islam, relations with Sunnis, especially the ultra-conservative Wahhabis of Saudi Arabia and Afghanistan (who are actually a great deal more unpleasant, violent, ultra-conservative and humorless) are frosty at best, and at times degenerate into outright war.
There are several sub-groupings of Shi'i, depending on who are recognized as legitimate lines of Imams. The two most important are the Seveners (who recognized seven legitimate Imams) who were in power in Fatimid Cairo, and the Twelvers (twelve Imams) who rule Iran presently. The Tweflth Imam of the Twelver line disappeared, creating quite a theological conundrum. The solution most Twelvers adopt is the belief that the Twelfth, or Hidden Imam is waiting, in an ageless state, to re-enter the world and set all injustices to right, a bit like the British King Arthur myth. Many Iranians believed that the Ayatollah Khomeini was the Hidden Imam returned.