want Ireland for the Irish, and the protestants want to be part of the UK.
As to Muslims...this is the best quick summation I found
The division between the two groups dates to the death of Muhammad in A.D. 632. The orthodox Sunni (or "example of the prophet") first emerged during a dispute about who would become the next caliph, the leader of the Islamic community.
According to Dudley Woodberry, professor of Islamic studies at the School of World Mission at Fuller Theological Seminary in Pasadena, the Sunni followed the Arabian practice of appointing a committee of elders to choose the leader. They elected Abu Bakr as caliph.
A second group, however, felt the rightful heir was Ali, cousin and son-in-law of Muhammad. Advocating that only direct descendents of the prophet should become caliph, Ali's supporters would become known as Shi'ites (or "the party of Ali"). They believe in a "divine light" that passes from Muhammad through his line to rightly guided imams. The Sunni do not believe in the divinity of Muslim leaders.
http://www.christianitytoday.com/ct/2003/117/31.0.html