The first is HEDGEWARS. This game is a clone of the Worms game with a nice twist on the weapons. The trailer video is a little long but worth watching. Hedgewars has a single player mode and a multiplayer/online mode.
My second pick is WARZONE 2100. Warzone 2100 is a RTS (Real Time Strategy) game. It reminds me of Command and Conquer or Starcraft. It is set in the post apocalyptic future and claims to have a unit build tree of over 400 different unit types. It has two single player modes (Skirmish and Campaign) and an online multiplayer mode. The video below is not mine but it gives you a good idea of what the game looks and plays like. The official trailer is a good intro to the game but doesn't show game play.
How do you get these fun and free games? Well this one is easy; Just open Ubuntu Software Center and search for HEDGEWARS and click on the INSTALL BUTTON, then search for WARZONE 2100 and click on the INSTALL BUTTON. Done! Now get your game on!