In the most elemental sense, improvisation is spontaneous creation.
Improvisational theatre is when the performers not only are responsible for the writing, editing, casting, directing, and acting but also responsible for accomplishing it all in the same moment. Improvisers simultaneously and spontaneously decide what role they will play, what they will say and do and how they will do it.
Improv comedy is the most popular form of improvisational theatre. Improv comedy is not stand up nor sketch comedy–which are usually written and prepared.
In improv comedy, the improvisers spontaneously perform games or stories which are often inspired by audience suggestions. You can see this kind of improv on TV in “Who’s Line is it Anyway?” and also at your local theatre or comedy club.