Another way of thinking of NP is it is the set of problems that can solved efficiently by a really good guesser.
The guesser essentially picks the accepting certificate out of the air (Non-deterministic Polynomial time). It can then convince itself that it is correct using a polynomial time algorithm. (Like a right-brain, left-brain sort of thing.)
Clearly this isn't a practically useful characterization: how could we build such a machine?
(Although, researchers have been recently entranced by the idea that superposition properties of quantum mechanics can simultaneously try out all guesses quickly. But, I don't think this panned out.)
