How to prove membership of NP
Here are several examples:


MAX-CLIQUE: The witness is a set of vertices of given size. The verifier checks that edges connect all pairs of vertices in the set.
SAT: The witness is a satisfying assignment. The verifier checks that it satisfies all clauses in the formula.
3COL: The witness is a 3-coloring of the graph. The verifier checks that all edges connect vertices of different color.
SUBSET-SUM: The witness is a subset of the set. The verifier checks that the subset sums to the target number.