For a graph, a maximum cut is a cut whose size is at least the size of any other cut. The problem of finding a maximum cut in a graph is known as the max-cut problem.
The problem can be stated simply as follows. One wants a subset S of the vertex set such that the number of edges between S and the complementary subset is as large as possible.
There is a more advanced version of the problem called weighted max-cut. In this version each edge has a real number, its weight, and the objective is to maximize not the number of edges but the total weight of the edges between S and its complement. The weighted max-cut problem is often, but not always, restricted to non-negative weights, because negative weights can change the nature of the problem.
For more details, see this