A follower of the Islam religion is called a Muslim, which in Arabic means "one who submits to God." Moslem is an alternate, older spelling that is now considered archaic.
In the past, followers of Islam were often referred to as Mohammedans or Mohametans in English works written by those who were not followers of Islam. This is culturally unacceptable to those who follow the Islamic religion. Mohammedan implies that the Muslim is a follower of Muhammed, the founder of Islam. Muslims prefer to keep the name emphasis on God and find all terms other than Muslim to be inconsistent with their beliefs.