Breadth-first search traverses a graph and in fact finds all paths from a starting node. Usually, BFS doesn't keep all paths, however. Instead, it updates a prededecessor function π to save the shortest path.
You can easily modify the algorithm so that π(n) doesn't only store one predecessor but a list of possible predecessors.
Dijkstra's algorithm applies more to weighted paths and it sounds like the poster was wanting to find all paths, not just the shortest.
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