One's aWoWance is the time allotted during a given period (typically by a guy's girlfriend, although female gamers could potentially experience this in reverse) for playing video games (as opposed to spending all one's time and energy trying to keep said significant other happy and thus putting out). The word itself derives from World of Warcraft (WoW), an extremely popular (and highly addictive) computer game that was/is known to draw players in for multiple hours and even entire days at a time; the term "aWoWance" can and does, however, apply to any computer or console game that inspires/requires focus and time to play properly.
Player 1: "Yo man, you gonna be on the Box later?"
Player 2: "Nah, bro-I can't tonight. I've already used up my aWoWance for the week, and I promised my girl I'd take her out to dinner and some ridiculous chick flick she really wants to see."
Player 1: "Bummer, dude."
Player 2: "Tell me about it..."