1. One who spends a lot of time playing games.
2. One who takes the playing of games very seriously.
3. One who does well at playing a certain game.
4. (Derogatory) One who plays the game for their own advantage; often used by players of role playing games to refer to players who they believe care more about character advancement than about role playing.
1. Being in a long session of playing games.
Bob is a powergamer who spends a lot of time planning about playing his game.
Bob is a Poker powergamer; he is really good at Poker.
Look at that 100th level Bob and his gear, man, is he a powergamer or what?
Bob is powergaming right now for 12 hours straight.
This is typically not a problem when dealing with games that encourage competitive playing, but swiftly becomes a problem when the objective of the game is not to defeat your opponenet, either real or perceived, but to co-operate and socialize with other players.
As such, power-gamers are the bane of certain role-playing games, social games and online games like MUDs and their ilk, and games where not everyone wants to make you their bitch.
However, certain games suit powergamers well -- games like Monopoly and the (now discontinued) Paranoia RPG setting by West End Games.
Power gamers are often wrongly known as munchkins. A munchkin gains power by cheating while a power gamer gains power by knowing and using the rules of the game to his advantage.
A) A person who frustratingly attempts to win by all means at tabletop role-playing games.
B) A person who seeks to accomplish everything possible within a videogame - meet all secret goals, get every item, unlock every secret, etc.
B) "You bought and upgraded every 200 cars in that game? Man, you're a powergamer."
Powergamers are different from twinks in that a powergamer generally seeks to have the most powerful character without regard to the rules, while a twink seeks to manipulate the rules to gain a powerful character.