Improvising on the piano involves several factors, but the most notable factor is the creation of a new melody while using the same chord progressions. In essence you are composing a new song, but the new song uses the same chords in the same order as the original song. Improvising is used in all genres, but it is most used perhaps in jazz, where a new melody is created several times in one song. Typically the lead sax or trumpet player will improvise for a period of time (on the same chord changes), then another member of the group will improvise for a period of time, and so on.

Arranging a song is somewhat different: you might keep the same melody, or alter it to some degree, but you might use different chords and different styles in the process. It’s a little bit confusing at this point, because the same type of thing could be happening in improvisaion. One musician will see it one way, and another musician another way, but that’s as it should be — it creates an endless flow of new and exciting music.