When you assume some fact at the very beginning of an argument, that's presupposition. If a book critic decides that a novel is intended for young adult readers before writing her review, she makes a presupposition.
When you presuppose, you make an assumption about something from the start, and presupposition is the action of presupposing. Your presupposition about how terrible a movie is going to be might color your actual experience of watching it, for example. The noun presupposition comes straight from the Medieval Latin praesuppositionem, which combines prae-, "before," and suppositio, "suppose."