Free includes pirated versions. Every single game released can be free, and lots of them are good.
free and Good: any game you can find a torrent for that you'd like. limited to singleplayer or completely terrible, buggy, cheat filled online
Good and Online: go to amazon/gamestop/walmart and find a multiplayer game
free and Online: any of the free korean mmos come to mind, also open source games. the problem is that many of these aren't very good