Improvisation means "freedom of ideas", so in music that would mean you are not limited to the traditional music nor to the written music. Pros often play from "fake books" -- a type of music notation (also called a lead sheet) that uses only the melody of the song along with chord symbols. Once a person knows chords, then they can play any note of a chord almost at random, plus scale notes and connecting passing tones that move between the chord notes.
For piano players, the right hand will usually play the improvised part using various piano notes, while the left hand plays the chord. But the improvised part is created from, and around, the chord. You'll learn how to use chords notes, neighboring notes, scale notes, and non-harmonic piano notes in your improvisations, and how each type functions in the overall scheme.
For more information on this facinating skill, go over to Piano Improvising Using Chords.