in CS we define a complexity class as a set of problems that have the same growth characteristic. that is for a problem size N, how long does it take in the worst case to find the solution for that problem.

one such class is the Polynomial class, or P, where the time to solution is some fixed exponent of N (like N^2, or 3).

the next big step is NP, which require a polynomial number of nondeterministic steps, whose solution can only be verified in polynomial time. usually solutions to NP problems are exponential in cost with respect to N (like 2^N), but thats not part of their definition.

problems in NP are generally identified by mapping them into a well known problem known to be in NP, where the mapping has to occur in polynomial time.

its an open question as to whether NP as a class can actually be solved in P time, but most people doubt that that is really the case.