Random graphs are an important and growing area of research in graph and probability theories, with applications in many fields such as computer science, biology, chemistry, physics, etc.  The notion of random graphs was introduced by Erdos and Renyi in their seminal 1959 paper.  This is known as the G(n,p) model, where edges between n vertices are assigned i.i.d. with probabilityp.  This model results in many interesting properties, but it fails to capture many of the properties associated with real-life graphs and networks such as heavy tailed degree distribution and high clustering coefficients.  To address these concerns there are many other models of random graphs which have been introduced.

A good starting point would be the 2006 paper of Joe Blitzstein and Persi Diaconis.