"Improv" for short, this is music made up on the spot to some extent. Like Improv Comedy, it is only for the brave entertainer. Sink or swim on the strength of instant performance.. it can be intimidating. Improv music grew out of jazz.
Much of jazz is patterned this way: the song starts and establishes a melody and chord progression that repeats, then the musicians take turns improvising solos over the chord progression, while the other players hold down the basic structure by "vamping" in the background. So the only improv is the soloist in this example. The soloist does his best to explore new ideas as he follows his own fancy over the backing music. He may stay close to home and play it safe, or he may go outside and really stretch it. Also, sometimes even the background structure is formless, so in that case the entire band is constantly improvising.
Improv is a highly evolved musical endeavor, and it can be horrible in the wrong hands. In the right hands, it can be a beautifully transcendant experience. The listener knows that he can hitch a ride on the musical thought process and take a little journey. The best improvisor is quick thinking, playful with ideas and unafraid to explore new territory, while remembering that he's got passengers.