Proc is a common term used primarily in game programming to refer to an event - a "procedure" - triggered under particular circumstances. A number Equipment articles have a Natrual Bonus to Proc a Spell. Armor have a chance to Proc when the specific item is Hit during combat. Weapons have a chance to Proc each time Damage is dealt with them. Shields have a chance to Proc when they are used to block.
Example: If a weapon has a 5% chance to Proc a Spell, then every time the weapon deals damage there is a 5% chance that the Spell will be cast on the same target.