The minimum cut is the cut with the smallest capacity. It represents the bottleneck in the network. If you remove all edges in the min-cut, no flow can reach the sink. The min-cut capacity equals the maximum flow value.
This is the max-flow min-cut theorem. Finding the min-cut is as easy as finding the max flow. Once you have the max flow, the min-cut is revealed by the residual graph. The min-cut edges are the bottleneck.