The Tribunal is a system in which players can regulate each other. When you enter the tribunal, via the League of Legends website, you are presented with a case. The case one of the thousands of daily reports submitted after games by the in game client. You get the chance to look over the case and decide if the reported summoner was really at fault for the situation. You can then choose to Pardon, Punish, or Pass the case based on your judgment. You can read more on the mechanics and rewards of the new system here.
That is, of course, if you can get to the website. At the moment when I try to access the Tribunal page, I'm met with Gragas.
-Tamat, Lead Community Manager, via the official forums.