Does NP mean verifiable in polynomial time or solvable in polynomial time? Is NP defined as verifiable in polynomial time, or solvable in polynomial time? Verifiable meaning that the solution can be checked in polynomial time, and solvable meaning that the solution can be found in polynomial time?
NP is a class of problems whose solution is verifiable in polynomial time on a deterministic Turing machine.
Alternatively, NP is a class of problems whose solution is solvable in polynomial time on a nondeterministic Turing machine. 

Please look here.