A music group is more known as a pop or rock band. It is usually composed of both singer(s) and musicians. They perform rock, pop or related music genres.
In most of the cases, a music group is formed on his own.
e.g. Linkin Park, Coldplay, Bon Jovi, Pet Shop Boys, Black Eyed Peas, Rolling Stones, Red Hot Chili Peppers, KISS, Queen, U2, Bee Gees, Led Zeppelin, The Who, etc.