Can someone explain why the MAX-CUT problem is in NP? Given an undirected graph $G = (V, E)$ and an integer $k$, is there a partition of the vertices into two (nonempty, nonoverlapping) subsets so that $k$ or more edges
have one end in each subset?

I'm a little confused on showing how the problem is NP, in terms of a certificate and certifier.
The certificate is a coloring of the vertices into red and blue (i.e., a partition into two sets). Given such a certificate, you can iterate through all the edges and count the number of edges whose endpoints have different colors. This count you can compare against $k$ and answer accordingly YES or NO.