Given a NP problem, a witness is an example proof that that problem is solvable (i.e. the answer is affirmative).

From Wikipedia, given a problem for which it is possible to find a solution only with nondeterministic polynomial time algorithms (NP problem), having a solution (and then just having to verify that the solution works) “speed up things a lot.” This is why it is important to have an answer to the NP problem/question.

this answer is called witness.