An equilibrium is a point towards which a system tends to move or a point which, once reached, tends to be stable.
The concept originates in physics. Imagine a room with a certain air temperature. You add a certain volume of warmer air and after a while the air in the room settles on a new higher temperature (all other things being, as always, equal).
In economics Adam Smith’s “invisible hand” provides an equlibrium. In a market economy, supply and demand make prices settle on a certain equilibrium point.

In terms of video games, two types of equilibria are important.

Equity equilibrium: A game may be constructed in a way which produces an a drift towards equality of score. Pool is a good example. The player with the most balls left has the best chances of actually landing a ball in a pocket. Thus, Pool is a game with an equity bias. All other things being equal, the score will tend to even out. Other games are equity neutral. In table tennis, for instance, winning a point does not increase or decrease one’s chances of winning the next ball. The game simply doesn’t care, if you will. Still other games have an inequity bias. Here, winning a round (or whatever) increases one’s chances of winning more. Chess is an example. Having many pieces left means having more power over the board. Once you’re ahead, you’re likely to stay ahead. This can all be understood in terms of feedback in the cybernetic sense of that word (see for instance Salen and Zimmerman’s Rules of Play on cybernetics).

Strategic equilibrium: A game may be constructed in a way which makes certain strategies highly succesful, sometimes given certain strategies played by the other player(s). In game theory terms a (Nash) equilibrium is a point where no player will unilaterally change his strategy – a point where both (or all) players are playing the best strategy given what the other player is playing. Game designers tend to dislike “best strategies” in this sense, as it implies that any choice facing the player is an uninteresting one (e.g. see Morris and Rollings’ discussion of strategic dominance). There are a number of challenges related to applying the idea to video games – chief among these are that it tends to miss the issue of skill (as it over-emphasises choice) and that it tends to miss the fact that playing video games is generally a very inductive process in which finding a “very good” strategy is a pleasant task in itself. Thus while clearly applicable and interesting, the concept is less than simple (or more than simple, if you prefer difficult things) as I will discuss in what-will-one-day be chapter 3 of my Dissertation. So now you know.