Basically, it's the inability of deterministic computers to solve non-deterministic problems in a reasonable amount of time. It has not been proven conclusively that this is impossible (do it and you win $1m), but no algorithm has been found. What's more, if one NP-complete problem can be solve in polynomial time, they all can be (and for non-solvable as well).
Famous examples of NP-complete problems are the traveling salesman and minesweeper.
Here's a good page on minesweeper.