If traditional gaming is generally about playing a game, metagaming is playing the game's system.
Essentially, it means using knowledge outside of the game's system/rule set to try to gain an advantage. In roleplaying games like Dungeons and Dragons, metagaming means utilizing information your character shouldn't have access to when making decisions, and is generally looked down upon as the lowest form of munchkinry. Other games, like Diplomacy and Shadow Hunters, actively encourage meta-gaming, since half the fun is trying to stay one step ahead of THAT SHITBAG MIKE WHO PROMISED HE WAS GOING TO HELP ME TAKE ANKARA HE PROMISED GODDAMN IT. Games aren't always separated from their respective metagame, and Mao is an example of a game that is sometimes considered a metagame.
Everyone who plays traditional and/or board games at all but the MOST casual of levels has probably experienced metagaming at some point; you've probably experienced it when your sleazy uncle was being That Guy while playing Monopoly during a family get-together.
Crafty metagaming is often hard to spot: doing a number of small but frustrating things during a game of Warhammer 40000 can make your opponent impatient and flustered, causing them to miss rules or skip moving units that would otherwise win them the game. The entire idea behind the "poker face" is rooted in metagaming. An example from video games would be counter-picking: instead of selecting a character, faction, equipment and/or even stage based on the game's internal balance, a player picks their stuff based on how they expect the opponent to play. It's not a mechanic within the core gameplay, but knowing your opponent as a person (and thus, their weaknesses, such as a tendency to beeline for the minigun pickup or get too drunk to play right by game 3) is important to doing well in 1v1 games like fighters or arena shooters, while predictability on your part is a weakness in itself.
The thing about metagaming is that any game-direct mechanics that are visible to players gives you the ability (and often the encouragement) to metagame, in a sense - as a player, you will probably have a lot of knowledge about how different classes and dice pools interact, and so to use your limited actions to your best advantage (or at least not to attempt things that are very unlikely). Only the most dickish of play groups would call this real metagaming. But it's a fine line between rational decision making and horribly abusing your knowledge of things your characters had no possible way to know about. If, for example, you happen to know than an NPC is statted in the book and you can easily kill him, and use that to inform your role playing to threaten or even kill him when he is presented as a hard ass you don't fuck with, that would be metagaming. Another example is a seasoned gamer facing a Beholder, even if the character has never seen one of the multi eyed things before, the player knows exactly what he's facing against. But at the same time that's just an inevitable consequence of playing a lot of D&D not of any nefariousness at work and not using that knowledge can be tricky. It's a complex concept really.
So it's like Powergaming?Edit
Except it's not. The line is fine between metagamers and powergamers, and both can certainly be the other at the same time, but they need not be one and the same. Powergamers eschew story and personal taste in order to be the most effective players, mechanically speaking. They optimize their strategy and choices in order to win the game. Metagamers use information or agency outside of the rules of the game in order to achieve an end goal, though that goal need not be winning. Powergamers can certainly metagame to win, but metagamers need not strive for victory in the game they play.
It should be noted that choosing mechanically superior items, skills, or traits based on statistics or numerical superiority is not metagaming trickery, that's powergaming (specifically min-maxing) shenanigans. Despite what many competitive video gamers may tell you, just because the developers didn't fix an overpowered mechanic doesn't mean that the powergaming community created a "metagame" around the game in question. They created an optimized strategy; distracting your opponents would be metagaming.
The Card GameEdit
There is also a card game that goes by the name Metagame.
It is played through making an argument for one particular piece of media, which is written on a card you select from your hand. The argument defends the media relative to a question card, drawn and played by the player who is first deemed a critic (by way of being the person whom last read a book). Each round, the players make an argument, and each round one defender has the best argument and one defender has the worst, determined by the critic. The best defender is given the chance to discard and draw from the media cards, to let them find something he might be able to better defend. The worst player is now a critic in an ever-growing pool of critics. The winner is the last defender standing.
Fun for your friends who know everything and like debating; not fun for the easily butthurt.