Proc
Proc is an abbreviation that refers a skill or item effect being activated when there is a percentage chance of activation. If a skill effect in the game has a chance to happen on a melee hit, then the percentage chance is the "proc rate".
Examples
If a Barbarian has invested skill points into Berserker State, the spell has a 6% "proc chance" when fighting enemies. This means there is a 6% chance the skill's active component will activate.
This is also true on items, where there is a chance of an effect going off on hit, and can also relate to spell casts, being hit, scoring a Critical Hit, Dodge or any other event in the game.
Background
The term "proc" is a programming term that refers to an event triggered under particular circumstances, or being "processed".
Proc is also sometimes short for "spec_proc" ("special procedure"), used by the original programmer of Circle-MUD, Jeremy Elson and possibly the original programmers of diku-MUD as well. These "special procedures" are functions assigned to objects, players, and locations to invoke a "special procedure" when a specific event occurs. These were used in a variety of ways in Circle-MUD, and might have originated the term.
When developers and players were talking about "special procedures" it was abbreviated to "proc". It evolved to a verb ("proced", "proc'd" or "procced"), indicating the procedure was activated.