The word "nondeterministic" suggests a method of generating potential solutions using some form of nondeterminism or "trial and error".

This may take exponential time as long as a potential solution can be verified in polynomial time.

NP is obviously a superset of P (polynomial time problems solvable by a deterministic Turing Machine in polynomial time) since a deterministic algorithm can be considered as a degenerate form of nondeterministic algorithm.

The question then arises: is NP equal to P?

I.e. can every problem in NP actually be solved in polynomial time?

Everyone's first guess is "no", but no one has managed to prove this; and some very clever people think the answer is "yes".

If a problem A is in NP and a polynomial time algorithm for A could also be used to solve problem B in polynomial time, then B is also in NP.