Depth First Search
Depth First Search is one of the main graph algorithms.
Depth First Search finds the lexicographical first path in the graph from a source vertex $u$ to each vertex. Depth First Search will also find the shortest paths in a tree (because there only exists one simple path), but on general graphs this is not the case.
The algorithm works in $O(m + n)$ time where $n$ is the number of vertices and $m$ is the number of edges.
Description of the algorithm
The idea behind DFS is to go as deep into the graph as possible, and backtrack once you are at a vertex without any unvisited adjacent vertices.
It is very easy to describe / implement the algorithm recursively: We start the search at one vertex. After visiting a vertex, we further perform a DFS for each adjacent vertex that we haven't visited before. This way we visit all vertices that are reachable from the starting vertex.
For more details check out the implementation.
Applications of Depth First Search

Find any path in the graph from source vertex $u$ to all vertices.

Find lexicographical first path in the graph from source $u$ to all vertices.

Check if a vertex in a tree is an ancestor of some other vertex:
At the beginning and end of each search call we remember the entry and exit "time" of each vertex. Now you can find the answer for any pair of vertices $(i, j)$ in $O(1)$: vertex $i$ is an ancestor of vertex $j$ if and only if $\text{entry}[i] < \text{entry}[j]$ and $\text{exit}[i] > \text{exit}[j]$.

Find the lowest common ancestor (LCA) of two vertices.

Topological sorting:
Run a series of depth first searches so as to visit each vertex exactly once in $O(n + m)$ time. The required topological ordering will be the vertices sorted in descending order of exit time.

Check whether a given graph is acyclic and find cycles in a graph. (As mentioned above by counting back edges in every connected components).

Find strongly connected components in a directed graph:
First do a topological sorting of the graph. Then transpose the graph and run another series of depth first searches in the order defined by the topological sort. For each DFS call the component created by it is a strongly connected component.

Find bridges in an undirected graph:
First convert the given graph into a directed graph by running a series of depth first searches and making each edge directed as we go through it, in the direction we went. Second, find the strongly connected components in this directed graph. Bridges are the edges whose ends belong to different strongly connected components.
Classification of edges of a graph
We can classify the edges using the entry and exit time of the end nodes $u$ and $v$ of the edges $(u,v)$. These classifications are often used for problems like finding bridges and finding articulation points.
We perform a DFS and classify the encountered edges using the following rules:
If $v$ is not visited:
 Tree Edge  If $v$ is visited after $u$ then edge $(u,v)$ is called a tree edge. In other words, if $v$ is visited for the first time and $u$ is currently being visited then $(u,v)$ is called tree edge. These edges form a DFS tree and hence the name tree edges.
If $v$ is visited before $u$:

Back edges  If $v$ is an ancestor of $u$, then the edge $(u,v)$ is a back edge. $v$ is an ancestor exactly if we already entered $v$, but not exited it yet. Back edges complete a cycle as there is a path from ancestor $v$ to descendant $u$ (in the recursion of DFS) and an edge from descendant $u$ to ancestor $v$ (back edge), thus a cycle is formed. Cycles can be detected using back edges.

Forward Edges  If $v$ is a descendant of $u$, then edge $(u, v)$ is a forward edge. In other words, if we already visited and exited $v$ and $\text{entry}[u] < \text{entry}[v]$ then the edge $(u,v)$ forms a forward edge.
 Cross Edges: if $v$ is neither an ancestor or descendant of $u$, then edge $(u, v)$ is a cross edge. In other words, if we already visited and exited $v$ and $\text{entry}[u] > \text{entry}[v]$ then $(u,v)$ is a cross edge.
Note: Forward edges and cross edges only exist in directed graphs.
Implementation
vector<vector<int>> adj; // graph represented as an adjacency list
int n; // number of vertices
vector<bool> visited;
void dfs(int v) {
visited[v] = true;
for (int u : adj[v]) {
if (!visited[u])
dfs(u);
}
}
Here is a generic implementation that additionally computes those:
vector<vector<int>> adj; // graph represented as an adjacency list
int n; // number of vertices
vector<int> color;
vector<int> time_in, time_out;
int dfs_timer = 0;
void dfs(int v) {
time_in[v] = dfs_timer++;
color[v] = 1;
for (int u : adj[v])
if (color[u] == 0)
dfs(u);
color[v] = 2;
time_out[v] = dfs_timer++;
}
Practice Problems
 SPOJ: ABCPATH
 SPOJ: EAGLE1
 Codeforces: Kefa and Park
 Timus:Werewolf
 Timus:Penguin Avia
 Timus:Two Teams
 SPOJ  Ada and Island
 UVA 657  The die is cast
 SPOJ  Sheep
 SPOJ  Path of the Rightenous Man
 SPOJ  Validate the Maze
 SPOJ  Ghosts having Fun
 Codeforces  Underground Lab
 DevSkill  Maze Tester (archived)
 DevSkill  Tourist (archived)
 Codeforces  Anton and Tree
 Codeforces  Transformation: From A to B
 Codeforces  One Way Reform
 Codeforces  Centroids
 Codeforces  Generate a String
 Codeforces  Broken Tree
 Codeforces  Dasha and Puzzle
 Codeforces  Making genome In Berland
 Codeforces  Road Improvement
 Codeforces  Garland
 Codeforces  Labeling Cities
 Codeforces  Send the Fool Futher!
 Codeforces  The tag Game
 Codeforces  Leha and Another game about graphs
 Codeforces  Shortest path problem
 Codeforces  Upgrading Tree
 Codeforces  From Y to Y
 Codeforces  Chemistry in Berland
 Codeforces  Wizards Tour
 Codeforces  Ring Road
 Codeforces  Mail Stamps
 Codeforces  Ant on the Tree
 SPOJ  Cactus
 SPOJ  Mixing Chemicals