This is a class of problems that are so difficult that even the best solutions cannot consistently determine their solutions in an efficient way.
Specifically, NP Complete problems can only possibly be solved in polynomial time using a nondeterministic Turing machine (a computer capable of making guesses and checking them in polynomial time).
See also Tractable / Intractable Problem
Because of the problem with finding the “best” solution, programs are often developed to find a usually reasonable solution.