Very simplified stated, it means that the solution to a specific instance of the problem takes at least polynomial time to solve - i.e. there can never be an algorithm that can solve it faster. In NP the time it is Non-polynomial - i.e. it takes such a **** long time that it is not really considered feasible for n>15 or such.
So if you have a specific stack of pancakes, err... I mean bagels... And want to turn all the sides without sugar downwards with the flip method from the article - it takes NP time to figure out how to do it best - also longer than polynomial time.
The cool thing is that NP problems can be converted to other NP problems in polynomial time and thus if just one of them can be solved in P time that all of them cab be solved in P time (P=NP) - we just don't know how yet....