The idea here, is if you have a solution, you can verify it in polynomial time by a deterministic Turing machine.
A problem is in NP if a nondeterministic Turing machine can find a solution in polynomial time. This hypothetical machine can make multiple guesses or follow multiple computational paths at once to find an answer quickly.
Hint
What is non-deterministic about something? Its because at a given point (in time, in execution, etc), for the same choice, there are multiple results.
Definition
Angelic non-determinism is when a non-deterministic algorithm chooses the favourable route of execution for a given desired result if the result is possible.
Demonic non-determinism is the opposite of this.
Arbitrary non-determinism is true non-determinism.
There is a subset of NP problems called NP Complete problems.