What is a Cut?

A cut in a graph is a partition of vertices into two sets $S$ and $T$ where $S \cup T = V$ and $S \cap T = \emptyset$. Every vertex is in exactly one set. An edge $(u, v)$ crosses the cut if $u \in S$ and $v \in T$ (or vice versa). The cut separates the graph into two disconnected parts.

If you physically removed all crossing edges, the graph would split into two components. Think of it as drawing a line through the graph. Edges that cross the line are cut edges. The two sides cannot communicate without crossing this boundary.