How does a Breadth-First Search work when looking for Shortest Path?
Technically, Breadth-first search (BFS) by itself does not let you find the shortest path, simply because BFS is not looking for a shortest path: BFS describes a strategy for searching a graph, but it does not say that you must search for anything in particular.
Dijkstra's algorithm adapts BFS to let you find single-source shortest paths.
In order to retrieve the shortest path from the origin to a node, you need to maintain two items for each node in the graph: its current shortest distance, and the preceding node in the shortest path. Initially all distances are set to infinity, and all predecessors are set to empty. In your example, you set A's distance to zero, and then proceed with the BFS. On each step you check if you can improve the distance of a descendant, i.e. the distance from the origin to the predecessor plus the length of the edge that you are exploring is less than the current best distance for the node in question. If you can improve the distance, set the new shortest path, and remember the predecessor through which that path has been acquired. When the BFS queue is empty, pick a node (in your example, it's E) and traverse its predecessors back to the origin. This would give you the shortest path.
If this sounds a bit confusing, wikipedia has a nice pseudocode section on the topic.
As pointed above, BFS can only be used to find shortest path in a graph if:
There are no loops
All edges have same weight or no weight.
To find the shortest path, all you have to do is start from the source and perform a breadth first search and stop when you find your destination Node. The only additional thing you need to do is have an array previous[n] which will store the previous node for every node visited. The previous of source can be null.
To print the path, simple loop through the previous[] array from source till you reach destination and print the nodes. DFS can also be used to find the shortest path in a graph under similar conditions.
However, if the graph is more complex, containing weighted edges and loops, then we need a more sophisticated version of BFS, i.e. Dijkstra's algorithm.