Probably one of the most famous freeware games of recent years, Cave Story is as good as its reputation says it is. An excellent platformer (with some adventure elements) with an excellent story that makes you care for these little pixel-creatures after some time.
You’re a little robot with no memory, awakening on a floating island where evil forces are at work to overtake, first the island, then the world. Cave Story is a higly polished game, as good as even the best commercial platformers, and it has some of the best pixel-art I’ve ever seen. The difficulty is moderate, only toward the final parts does the game becomes extremely difficult (I never managed to beat anything beyond the last village without cheating) and the secret hell level is in a class of its own (I doubt a lot of people can beat this without cheating).