Briefly (and wrongly, but it will do for a slashdot reply), NP complete problems require the computer to actually calculate all of the possible solutions and see which one is best rather than than using an algorithmic shortcut. For every extra item that has to be computed the complexity increases by quite a large amount relative to the difficulty of the smaller problem. The traveling salesman problem is the classic example - adding another town to visit means you have to recalculate all the routes because there just might be a better solution using the new town in any part of the route.
Anyway, there's nothing impossible about NP complete problems - the issue is that they get very hard very quickly as you make the problem space bigger. Quickly the time required to find the best solution (not just a good one) would take longer than the remaining lifetime of the universe.
One of the tricks with NP complete problems is that if you're looking for a merely good solution, not the absolute best, humans can often use some heuristic tricks to home in on a good solution quickly while a dumb computer algorithm would still be chugging away looking exhaustively for the best solution. Studying the kinds of guesses that humans make in these situations is a notable area of study in artificial intelligence.
In summary, a computer will kick you ass at minesweeper, but it still won't be able to solve a 10^14 x 10^14 board before the end of time.