The minimum - cut is the cut with the smallest capacity among all - cuts. It is the cheapest way to disconnect from . There might be many different cuts, but this one has the lowest cost. You are looking for the bottleneck: which edges, when removed, stop all flow with the least total capacity lost?
This is the network's vulnerability point. This cut represents the weakest point in the network between and . Attack this cut, and you break the connection most efficiently.