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).
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.)
