If you've never heard or understood either of these games, you're not alone. On paper, MOBAs, or Multiplayer Online Battle Arenas, make sense. Largely similar, these games are reduced real-time strategy games in which you guide an avatar (like one of the many portraits you see above for Dota 2) to attack and defend lanes of combat flowing from the enemy's base. You'll level up, gain powers, and move your avatar across the map to help your teammates while micromanaging combat or destroying structures to help your cause. The learning curve for both is also sheer, so if you haven't bitten the apple already, you'll have a fun time figuring out which skills to acquire and which abilities to purchase (and when!) over plenty of time.
But hey, they're free!