In jazz, a vamp is simply a repeating musical figure or accompaniment (Corozine 2002, p.124). The equivalent in classical music would be an ostinato. A background vamp provides a performer, or perhaps the pianist's right hand, a harmonic framework upon which to improvise. A vamp often acts as a springboard at the opening of an improvisation.
Similarly, in musical theater, a vamp is a figure of one or two measures which the orchestra repeats during dialogue or stage business. Here the purpose of a vamp is to allow the singers as much time to prepare for the song or the next verse as is necessary, without either requiring the music to pause until the singers are ready or requiring the action on stage to be carefully synchronized with music of a fixed length. (Courtesy of Wikipedia)
A vamp is a simple one- or two-bar chord progression that is repeated to create a song introduction or ending. The term "vamp till ready" means that the vamp sequence is repeated until the enterance of the singer or soloist. In the 1960s and 1970s, vamps were used instead of more complex chord sequences as verse and chorus progressions. Three great vamp examples are shown below in the key of C.