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).
Edit:
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.)
