Though it's short, The Stanley Parable is a free Source mod (so you just need a Source game like, oh say, TF2). You owe it to yourself to play this game. Your mind will be blown, your head will explode, your socks will be knocked off, your face will be melted. Do not read anything about the game before playing it (spoilers will pretty much ruin it for you). Again, if there is one game in this thread that you play, play that. Seriously, I cannot say enough good things about this game.
Though the playerbases are pretty small at this point, Pirates Vikings and Knights II and Dystopia are also pretty fun Source mods (multiplayer obviously).
Cave Story is also an absolute must. A modern retro masterpiece.
League of Legends is a great free multiplayer game (I personally like it better than the other MOBA games like Heroes of Newerth and it certainly has a bigger playerbase, but your opinions may differ).
Also, play Passage. It's free, exceedingly short, and distressingly poignant.
Edit: I almost feel bad suggesting this, but if you ever feel like playing a truly Nintendo Hard free game, I Wanna Be the Guy. Understand that this game is an exceedingly masochistic experience.