What's maximal clique? I'm unable to understand what maximal clique is. I mean how a clique can't be extended by a node and remain a clique? If I add a node and then I connect this node to every other nodes in the clique, then it got extended and remained a clique!
 A: The phrase "maximal clique" is usually used in terms of a subgraph of a given graph $G$. So a subgraph $H$ of a graph $G$ is a maximal clique in $G$ if $H$ is isomorphic to a complete graph and there is no vertex $v \in V(G)\backslash V(H)$ so that $v$ is adjacent to each vertex of $H$.
In other words, a subgraph $H$ of a graph $G$ is a maximal clique in $G$ if $H$ is a clique (there is an edge between every pair of vertices in $H$) and there is no vertex in $G$ but not in $H$ that sends an edge to every vertex of $H$. So you couldn't create a bigger clique in $G$ by adding another vertex to $H$.

The red subgraph of the first graph is not a clique because there are two vertices in it not connected by an edge. The red subgraph of the second graph is a clique, but because there is a vertex in the larger graph connected to all 3 vertices in the subgraph, it is not a maximal clique. The red subgraph of the third graph is a maximal clique because it is a clique, and the last vertex not included in the subgraph does not send an edge to every vertex in the subgraph. The red subgraph of the fourth graph is a maximal clique because it is a clique, and neither of the vertices not included in the subgraph send an edge to every vertex in the subgraph. Note that the third and fourth red subgraphs are both maximal cliques even though the the third red subgraph is larger. 
