A Graph (aka network, diagram) is basically a tuple of two elements, namely a
set of vertices and edges.
G = (V,E)
Basically a graph states that something is related to something else.
There are several types of graphs. A graph can be undirected and directed (digraph). If one can find more than one relationship between the same pair of nodes than one has a multigraph. In a hypergraph a relationships is able to connect more than two nodes. Property Graphs are directed, attributed and multi-relational.
Data is getting more and more connected (from text documents, to Wikis, to Onotologies, to Folksonomies, etc.), more semistructured (decentralization of Content generation) and more complex (Social Networks, semantic trending, etc.)
Graphs have numerous applications: