Any person that just doesn't get 'it', and isn't aware of how sad it is that they don't get 'it'. A mildly derogatory term used by a member of some elite group to describe people outside of the group.
The 'it' varies widely from group to group; for nerds, normies are people that can't tell kinetic energy apart from their dicks; for geeks, anyone that scoffs at their D&D capes that they wear when they go to Burger-King; hipsters, anyone wearing and American Eagle shirt...
Essentially, 'normies' are the 'muggles' of the world, if the word 'muggles' didn't carry so much harry potter baggage.
Shit nuggets, I miss college. Being surrounded by normies all the time is driving me nuts.