Proc is gaming slang indicating a triggered event. It originally comes from CircleMUD, and was carried over into World of Warcraft and other MMORPGs, and later into card-based games such as Magic: The Gathering.
Originally, proc was simply a shortening of the spec_proc (short for "special process") command in CircleMUD, a function that was used to trigger certain events when certain actions were taken. Proc quickly became the term of choice to refer to special events, and soon became a verb (to proc; proc'd), meaning to trigger a special event.
This term took off among World of Warcraft players, and with this a new etymology appeared as well -- many people will tell you that proc is short for "Programmed Random OCcurence". At the same time, the meaning expanded to include events that had a specific probability of triggering (Chance on Use or Chance on Hit), e.g., events that have only a 10% chance of triggering.
Proc has since spread to other games, and its meaning changes slightly from context to context. However, in most cases it remains largely a synonym for 'trigger'.