I have just completed the game, and it says to try Dark World. What is that and how do I get there?
The Dark World are just the same levels a few magnitudes more difficult.
To switch between Dark and Light World, you just have to press X on your gamepad or Shift on your keyboard in the level select screen (see an example on Wikia).
Note that the Dark World levels becomes available after you beat the corresponding Light World levels on A+. (Thx to shadow for pointing that out)
To clarify what I mean by "just the same levels a few magnitudes more difficult", I refer you to the blog of the creators of SMB. (The images are down somehow, so I linked to a web archived version.)