What are Achievements?
Achievements are benchmarks within the game that set goals for what you are to do. For example, once you beat Kang-Kung, you get "the New Boss" achievement, which awards you with a new title. As you progress further into the game, these achievements will eventually become harder and harder to accomplish, and will therefore award more Boss Points.
Achievements can be found when looking at Dungeon Boss's main option menu, where you see your level and whatnot. When tapping on "Recent Achievements" you will find four tabs that organize the Achievements; Campaign, Player, PvP, and Heroes, all with achievements that which associate with the tabs.