a certificate
A certificate is the answer key slipped into the back of the textbook. The problem in front of you might be a brutal search, but the certificate is the short note that says 'here is the answer, and here is exactly why it works'. You did not have to find it, but armed with it you can confirm the result in moments. In complexity theory a certificate is the piece of side information that turns a hard search into a quick check.
Concretely, a certificate is an extra string c handed to a verifier alongside the input x. It is the proposed evidence that the answer to x is 'yes'. The defining requirement, for NP, is that the certificate be SHORT: its length must be bounded by a polynomial in the length of x, so the verifier can read it within its polynomial time budget. A good way to picture a certificate is 'the thing you would point to if a skeptic demanded proof of a yes-answer'. For 'is this graph 3-colourable?', the certificate is an actual colouring of the vertices with three colours. For 'does this Sudoku have a solution?', it is a filled grid.
Certificates only exist for 'yes' instances, and only one of them needs to work. The definition of NP says 'there EXISTS a certificate the verifier accepts'; if the answer is 'no', every candidate certificate fails. This asymmetry is the whole reason NP and its complement co-NP can differ. It is also why certificates are so useful as a design tool. To show a problem is in NP, you decide what a 'yes' looks like, argue that evidence can always be written down briefly, and confirm a checker can validate it fast. Note that a certificate certifies an answer; it does not have to reveal how the answer was found.
For the Hamiltonian path problem, the certificate is an ordering of the vertices, say v3, v1, v4, v2. The verifier checks two things in polynomial time: that the list is a permutation of all vertices, and that consecutive vertices are joined by edges. If both hold, this ordering certifies a 'yes'.
A certificate is short evidence for a yes-answer that a verifier can check quickly; no certificate works for a no-instance.
The certificate must be polynomially short. 'The full table of all solutions' is not a valid NP certificate because writing it down can itself be exponential in the input size.