Your question is equivalent to finding a minimum s-t cut in the graph, since this cut gives the smallest set of edges that, if removed, disconnect s and t. This is the same as saying that every path goes through some edge in the minimum cut.
There are many algorithms for finding minimum s-t cuts. For example, the max-flow min-cut theorem states that the value of a max flow from s to t (if each edge has unit capacity) has the same flow as the number of edges in the min s-t cut. Consequently, any max-flow algorithm, such as Ford-Fulkerson or Edmonds-Karp, can be used to directly compute the cost of a min cut. From there, it's easy to recover the min cut by finding all edges reachable from s in the residual graph and taking all edges that have one endpoint in this set and another endpoint in the complement.
Hope this helps!