In a lot of math, the prefix "pre-" is added to a structure you want to talk about, but you don't quite have, but you have a general construction to get you there. Given a preorder, there is a general construction which turns it into a partial order. Given a presheaf, there is a general construction which turns it into a sheaf (called sheafification :P). That's why its "pre-" and not "semi-" or some other "half", because its not quite there, but you know it will get there eventually!