Finding the absolute maximum amount of water you can push through a network of pipes.
In a Flow Network, edges have a maximum capacity. The Max-Flow Min-Cut theorem is one of the most beautiful results in computer science: the maximum flow from a Source to a Sink is exactly equal to the capacity of the smallest bottleneck (the Minimum Cut) that separates them.
Edmonds-Karp (Ford-Fulkerson using BFS) finds the max flow by repeatedly finding an "augmenting path" from source to sink with available capacity, pushing flow along it, and updating "residual" capacities (which allow the algorithm to "undo" bad flows by pushing water backwards).
The magic is the residual edge. If we push 2 units from A to B, we add a back-edge from B to A with capacity 2. This means later, if we find a better path, we can "push back" that 2 units of flow, effectively cancelling the earlier decision!