Mūmin
Mūmin (feminine: Mūmina) is an Arabic Islamic term, frequently referenced in the Quran, meaning "a sincere believer". It denotes a person who has complete submission to the Will of Allah and has faith firmly established in his heart, i.e. a "faithful Muslim".
Also, it is used as a name and one of the 99 Names of God in Islam.
In the Quran
The Quran states:
O you who believe!
Believe in Allah, and His Messenger, and the Book (the Quran) which He has sent down to His Messenger, and the Scripture which He sent down to those before (him),
and whosoever disbelieves in Allah, His Angels, His Books, His Messengers, and the Last Day, then indeed he has strayed far away. /an-Nisā 4:136/
This verse addresses the believers, exhorting them to believe, implying multiple stages of belief.
Difference between Muslim and Mūmin
The term Mūmin is the preferred term used in the Qur'an to describe monotheistic believers.
The following verse makes a distinction between a Muslim and a believer:
The desert Arabs say, "We have believed."
Say, "You have not [yet] believed; but say [instead], 'We have submitted,' for faith has not yet entered your hearts.
And if you obey Allah and His Messenger, He will not deprive you from your deeds of anything. Indeed, Allah is Forgiving and Merciful." /al-Ḥujurāt 49:14/
Mūmins are said to be saved by Allah on the Day of Judgement, but it is not granted for everyone who only calls themselves Muslims, but may turn out to be Fāsiqs or Munāfiqūn.