Proc or proccing means occurring of a random event or effect, often related to characters' actions such as critical hits or effects provided by upgrade components such as Sigil of Earth or Rune of Rage.
In Guild Wars 2, proccing most commonly refers to occurring of special effects of upgrade components or traits based on random chance. It can also sometimes refer to any effect which occurs when the character is attacked, attacks or is in a certain state such as affected by a certain boon or condition. In the former case, an example would be Superior Sigil of Strength which has 30% chance of applying might boon on the user on their critical hits. For the latter kind of definition of proccing, a trait such as Hidden Flask procs when the user takes damage while below 75% of their maximum health.
The most popular definition of proc or proccing is "Programmed Random Occurrence".