The following is a comprehensive answer to what you are asking.

In simple terms:

Breadth First Search (BFS) algorithm, from its name "Breadth", discovers all the neighbours of a node through the out edges of the node then it discovers the unvisited neighbours of the previously mentioned neighbours through their out edges and so forth, till all the nodes reachable from the origional source are visited (we can continue and take another origional source if there are remaining unvisited nodes and so forth). That's why it can be used to find the shortest path (if there is any) from a node (origional source) to another node if the weights of the edges are uniform.

Depth First Search (DFS) algorithm, from its name "Depth", discovers the unvisited neighbours of the most recently discovered node x through its out edges. If there is no unvisited neighbour from the node x, the algorithm backtracks to discover the unvisited neighbours of the node (through its out edges) from which node x was discovered, and so forth, till all the nodes reachable from the origional source are visited (we can continue and take another origional source if there are remaining unvisited nodes and so forth).

Both BFS and DFS can be incomplete. For example if the branching factor of a node is infinite, or very big for the resources (memory) to support (e.g. when storing the nodes to be discovered next), then BFS is not complete even though the searched key can be at a distance of few edges from the origional source. This infinite branching factor can be because of infinite choices (neighbouring nodes) from a given node to discover. If the depth is infinite, or very big for the resources (memory) to support (e.g. when storing the nodes to be discovered next), then DFS is not complete even though the searched key can be the third neighbor of the origional source. This infinite depth can be because of a situation where there is, for every node the algorithm discovers, at least a new choice (neighbouring node) that is unvisited before.

Therefore, we can conclude when to use the BFS and DFS. Suppose we are dealing with a manageable limited branching factor and a manageable limited depth. If the searched node is shallow i.e. reachable after some edges from the origional source, then it is better to use BFS. On the other hand, if the searched node is deep i.e. reachable after a lot of edges from the origional source, then it is better to use DFS.

For example, in a social network if we want to search for people who have similar interests of a specific person, we can apply BFS from this person as an origional source, because mostly these people will be his direct friends or friends of friends i.e. one or two edges far. On the other hand, if we want to search for people who have completely different interests of a specific person, we can apply DFS from this person as an origional source, because mostly these people will be very far from him i.e. friend of friend of friend.... i.e. too many edges far.

Applications of BFS and DFS can vary also because of the mechanism of searching in each one. For example, we can use either BFS (assuming the branching factor is manageable) or DFS (assuming the depth is manageable) when we just want to check the reachability from one node to another having no information where that node can be. Also both of them can solve same tasks like topological sorting of a graph (if it has). BFS can be used to find the shortest path, with unit weight edges, from a node (origional source) to another. Whereas, DFS can be used to exhaust all the choices because of its nature of going in depth, like discovering the longest path between two nodes in an acyclic graph. Also DFS, can be used for cycle detection in a graph.

In the end if we have infinite depth and infinite branching factor, we can use Iterative Deepening Search (IDS).