Balance is a tricky subject to talk about. Most people think balance is about fairness. It's not. Balance is about choice.
What is Balance?
Balance is the factor that maximizes the player's meaningful choices within the context of the rules of the game.
Got that? Yeah, me neither, so let's break it down a little.
Why do we have rules? We have rules in order to provide the player with a system within which they can make interesting choices. Imbalance weights a player's choice to a degree that it no longer becomes interesting. The classic example of this is the RPG item that you never unequip. When you find an item that is so good that every other item you could equip in its place becomes worthless, all the interesting choices that you should have had to make about equipping that slot become moot.
Whew, maybe that was still a bit much. I think we need some examples. First let's look at Final Fantasy VII. Once a player gets Quadra Magic, all the choice is taken out of fights. The only question one has to ask oneself when fighting becomes, "Do I really want to sit through 10 minutes of Knights of the Round?" This doesn't make the game too easy or too hard, but it leaves it out of balance.
Now let's look at World of Warcraft. Grey items in Warcraft can be considered out of balance. Why? Because no meaningful choice surrounds them; they are always sold to a vendor. If you think of them as "equipment" rather than cash, they are completely out of balance with everything else in the game. (Yes, they serve to intensify the jackpot feeling you get when you find a magic item, but looking at them simply as items they are unbalanced.)
Why is balance so often associated with fairness? When things are seriously out of balance one of two things occur:
None of the player's choices have meaning because all of her choices lose.
None of the player's choices have meaning because all of her choices win.
In case 1, the game feels too hard. In case 2, the game feels too easy. This doesn't mean we can't make balanced games that are very hard, like Ninja Gaiden, or fundamentally easy, like Pokemon. It just means that we have to be careful about how we go about it.
A dominant strategy is something to avoid at all cost. A dominant strategy is a method of play that is so good, it makes all other methods of play irrelevant. Tic-Tac-Toe is the best example of this. There's a strategy to Tic-Tac-Toe that will assure that you never lose. Once you know this strategy, it's all you ever do. Tic-Tac-Toe thus becomes a mindless activity rather than a game. (In fact, so much so that someone has trained chickens to do it in Las Vegas.)
Fair but out of Balance
Over the years many game designers have tried to mathematically balance their games. It's what we're wont to do. It seems so elegant, so simple ... but there's a rub.
Mathematically balanced games are always fair, but they're almost never balanced. Let's take a quick look at what mathematically balancing a game means.
To mathematically balance a game, you simply assign point values to all the game elements (this is easiest to think of in terms of stats in an RPG, though many other game types use mathematical balance) and make sure that any given game object (a unit, an item) has the correct point value relationship to all the other game objects. The problem with this type of thinking is that it hems you in as a designer and obscures the question of balance with the smug realization that your game is eminently fair.
Take for example Oblivion. In Oblivion, the enemies are scaled to the player's level and desired difficulty level. While this ensures fairness, it strips away a significant amount of player choice (not to mention design flexibility).
Ignoring for a moment the obvious damage this scaling does to the standard reward system of an RPG (that is, a clear increase in power), let's look at some of the choices it takes away from the player.
The player never has to decide if he is ready for the next challenge.
The order in which the player chooses to face challenges becomes inconsequential. (I believe this is why the developers chose this system. As Oblivion is an open world game, the scaling system prevents the unfair but not necessarily unbalanced moment when the player wanders into an area that is much too high level for him and is destroyed. MMOs offer an open world experience without utilizing such a system, though, in part, this is due to their "massive" nature.)
The player can't choose between high risk-high reward adventuring and low risk-low reward adventure, only high or low difficulty.
Perfect mathematical balance takes agency away from the player because it involves the tacit assumption that the player always wants things to be fair. In fact, this is rarely the case. Rather, the player always wants the game to be interesting. (For a case study on mathematical balance try PoxNora; all its math is exposed.)