Intuition behind min cut in a flow network? Whether it's baseball elimination or project selection
Given a weighted directed graph and two nodes $s,t$, an $s$-$t$ cut is a set of edges, upon whose removal $t$ is not reachable from $s$. The weight of the cut is the total weight of edges removed. A minimum $s$-$t$ cut is one of minimum weight.

A similar definition works for undirected graphs and in the special case in which the graph is unweighted, in which all edges have unit weight.

If we think of a weighted digraph as a flow network with source $s$ and sink $t$, then any $s$-$t$ cut gives a restriction on the weight of any flow in the network, which must be at most the weight of the cut. The min cut max flow theorems states that the weight of a minimum $s$-$t$ cut coincides with the weight of the maximum flow. However, a cut and a flow are two very different things.

There's a related definition of cut which works just for undirected graphs.
A cut in an undirected graph is a set of edges whose removal disconnects the graph. (Any $s$-$t$ cut is a cut.) A minimum cut is one of minimum total weight (or, in the unweighted case, of minimum cardinality).

One reason that cuts are so powerful is that they can be found efficiently.