Development
The petri net was first created by Carl Adam Petri in his PhD thesis in 1962.  With subsequent developments generalising the petri net, and allowing a wider variety of applications, the petri net is now a commonly used tool.

Definition
The petri net is a directed bipartite graph where each node is a member of a set of transitions, or a set of places.  The definition of a petri net must show the set of places, the set of transitions, and the bag of arcs between the two.  This can be represented by C=(P,T,I,O) where P is the set of places, T is the set of transitions, I is the bag of inputs from a place pi to a transition tj, and O is the bag of outputs from the transition tj to the place pi.
 

Links:
These sites provide a good general overview of the Petri net, for anyone who's interested.
 Petri Nets Bibliography
 The World of Petri Nets