Metagaming can be defined as any aspect of strategy that involves thinking about what your opponent is thinking.
I'm going to assume that you've played the game Rock-Paper-Scissors (RPS). The "game" of RPS is almost entirely a metagame. You are trying to figure out what your opponent will throw so that you can throw whatever will beat it. This is the simplest form of a metagame, but it's a great way to introduce the concept of Levels of Thinking.
- Level 0 Thinker
"I'm going to throw rock, nothing beats rock."
- Level 1 Thinker
"I know my opponent is going to throw rock, so I'm going to throw paper."
- Level 2 Thinker
"I know my opponent thinks I will throw rock, and so he's going to throw paper, so I'll surprise him and throw scissors."
Adding additional levels will continue to change what you throw based on about what you think your opponent thinks you think your opponent thinks you will throw. Follow that? Good.
This general thought process is used in nearly every game, and for most people, it doesn't go past the first or the second level of thinking, because anything beyond that requires that you have great judgement on the true level of thinking of your opponent. In the case of RPS, the choice of a third level thinker is the same choice that a level zero thinker would choose, which illustrates how there are times where genius can look like ignorance and vice-versa.
In fact, if you judge your opponent as a 3rd level thinker who is going to throw rock, so you should throw paper, and he is really a 2nd level thinker who is going to throw scissors, you've obviously made the worst choice possible because scissors beats paper. Since RPS only has three selections to choose from, the levels of thinking are overly simplified as they will circle themselves very quickly, but they illustrate well both the benefits of higher level thinking, as well as the pitfalls of poor judgement.
The poker variant of No-Limit Hold-em is probably one of the most major, yet recent, example where a new metagame has greatly changed and evolved in the last 10 years. If you've watched any recent poker tournaments on TV, you've no doubt seen people re-raising a player with some very questionable hands. This is due to the evolution of the poker metagame and the players understanding that the cards don't matter if you can get your opponent to fold. It used to be that a re-raise meant that you had Kings or Aces in your hand. Eventually, players started to use that line of thinking to their advantage, being able to represent those hands far more often than they were actually dealt them. When their opponents realized this fact, they had to adjust their play accordingly, and push back with hands weaker than they used to.
In the end, all forms of competition will benefit from higher levels of thinking. From video games to board games to sports to poker and even business, knowing what your opponent is going to do, or knowing what they think you are going to do is going to give you a huge advantage in your decision-making.