Confirmation in Mafia can refer to one of two things. The first is an acknowledgment of receipt and understanding of one's role PM. Such a confirm is usually sent out by a player at the beginning of the game after getting his or her role and asking all relevant questions. Some mods will have confirms made in the thread, and others will have them made via PM. This can also depend on if the game is a Day Start or Night Start game.
The other use is to describe a means by which some aspect about a player's role, alignment, or actions can be guaranteed. This can be brought about by many means, including Cop investigations, Flavor Cops, role flavor, and so on. While it is almost impossible to be 100% cleared in Mafia, you can get pretty close (a true/perfect confirmation is sometimes called "mod-confirmed"). A person that is confirmed innocent is also sometimes referred to as "cleared".
See also Confirm Vote.