Diablo III has multiple difficulty levels that you must play through to fully experience the game.
Diablo I and Diablo II both had Normal, Nightmare, and Hell difficulties. Diablo III introduces Inferno difficulty.
To progress from Nightmare into Hell requires that you must kill Diablo for the quest in Nightmare, and be at least level 50.
To enter Inferno difficulty, you must be at least level 60 (the current maximum level of the game), and have completed the quest to kill Diablo in Hell difficulty.