Achievements are awarded for playing regular aspects of the game, such as running a certain distance or killing bosses. You'll also need to Legend to unlock some of the later achievements. Some are found in Story Mode, too.
Achievements often give a gold or items bonus, but they can also unlock Weapons, Armor, and Accessories for later purchase.
To check your Achievements go to the middle left of the screen and click on the blue button that has a cup on it.
Images of all achievements, courtesy of Cain.