Variation form is just another way of developing a theme or musical idea.
At the beginning of the movement or musical section, a theme is presented, usually in its unadulterated form.
A series of variations follow - each variation will elaborate on the original through changes in the rhythm, added ornamentation or extra notes. (These are just some of the techniques)
Very often, the theme and the subsequent variations are divided into two parts which are both repeated (ie: AABB)
Beethoven loved the variation form and the last movement of the Eroica (his 3rd symphony) is a great example.