You will earn achievements when you hit various milestones in the game. When you reach an achievement you'll be given a variety of rewards for completing it, including Experience, Gold, Skill Points, and even bonuses that make your character heal, attack, or gain energy faster!
There are currently hundreds of Achievements for you to reach! To see which achievements you have, and which ones you have yet to unlock, go to your Profile, and select the Achievements tab.
Once you're on the Achievements page you can pick one of seven categories to see achievements from every aspect of your game.