Sunni vs Shiite Islam
When Ali and Husayn were murdered, a split known as a "schism" formed within Islam. This schism resulted in two seperate sects of Islam, the Sunnis being the majority and the Shi'ites being the minority. There are many distinctions between the two sects, this is only a summary of some significant differences.
Shi'ites believe the caliphate can only be held by decendants of Muhammed through his daughter Fatimah. Muhammed left behind no instructions on the matter of leaders in the Muslim world, so there is no way to tell for sure which method of choosing a leader is best. In Sunni Islam anyone who is pious, well-liked and accepted as a devout muslim can potentially be a caliph, whereas the Shi'ites developed the belief that the descendant (Imam) who was their leader was saintly and divinely appointed.
The Shi'ites and the Sunnis have the same Qur'an... however, the difference is that Sunnis take the Qur'an for meaning exactly what it says, whereas the Shi'ites often have very different interpretations of the text. Generally the Shi'ites feel that the Qur'an is not the "final" word but rather believe that their Imams can expand on it.
The difference in prayer is basically a result of the variation in interpretation of the traditional beliefs. The Sunnis do not choose to reinterpret, change, or modify the traditional teachings and so their prayer reflects this. The Shi'ites on the other hand tend to praise not only Allah but his followers, their religious leaders, like Hussein. Also, the Imams have the ability to change laws and doctrines and because of this they have chosen to change the call to prayer and the posture used in prayer for their own reasons.