Trophies are a form of unlockable available in compatible games. The feature was introduces in Firmware 2.40. You can unlock trophies by completing a certain goal in a game. The number of trophies in a game depend on how big the game is: small or big. There are four types of trophies you can collect:
The platinum trophy can only be obtained once you collect every other trophy in the game. The trophies equal to points, which determines your level on your PlayStation Network Profile. So, the more trophies you have unlocked, the highter your level. You can view your own trophies by going to [Game] > [Trophy Collection]. You can also view a friend’s trophies by clicking on their name from the [Friends] menu.
Although the trophies can’t be used to redeem any rewards, they are great for bragging rights, sorta like a way of showing your friends that your better than them at a certain game :).