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.