I think it's easiest to set up a definition of game balance. A balanced game is one in which no given character option or set of options (i.e. classes, skills, powers, spells, special abilities, merits, flaws, etc.) has an unfair advantage over any other set. Most gamers I know would agree with that definition, although we might argue over the particular wording, but the crux of any disagreement would be the word, "unfair." What makes an option fair or unfair?
Accessibility is a big factor. If I'm playing D&D and all the spells in the Tome of Ultimate Badassery (Vol. 6) are more powerful than the spells in the main book, then any player who buys ToUB6 has an innate advantage over players who just buy the core book. If access to this supplement is limited, such as it being a limited print run, expensive, or some players just not having the time to comb through each supplement to find the best options, then those who do have access have an unfair advantage because they have something that other players do not.
Campaigns that allow character development between sessions, such as larps with blue-booking sessions or living campaigns that give free Advantages or treasures in exchange for player-written fictions can be considered unfair and thus unbalanced. The former gives advantage to players with a great deal of extra time while the latter gives advantage to players with greater skill at writing.
Another unfair advantage is if access is available to all players but not their characters. For example, in Legend of the Five Rings the schools of the Great Clans are significantly more powerful than those of the Minor Clans (though this was even more egregious in 1st edition). A player who wants the roleplaying challenge of portraying a minor clan samurai is thus also stuck with a mechanical penalty.
Still another unfair advantage is when some mechanical options are so powerful or low cost that any player who doesn't use those options is placed at a disadvantage (a dominant strategy). This is particularly common in games that use merits and flaws/advantages and disadvantages, which give incentive to players to take the most powerful merits with the lowest costs that they can while taking flaws that give many points but at a relatively low or rarely-occurring penalty. An example would be the Unbondable merit in Vampire: The Masquerade, which for only a few character points negates one of the most powerful tools in the Kindred's arsenal.
Now there are several responses to an unbalanced game which I'm going to briefly touch on.
Fix it: This is one of the most common solutions, wherein the GM tries to adjust the mechanics so that the advantage is no longer unfair. This may mean increasing the cost of a certain option, making mechanics universally available, or otherwise changing the situation so that no option is favored over the others. The problem is that this fine-tuning requires constant adjustment and GM awareness, and due to Chaos Theory may have unforeseen consequences and end up unbalancing other areas of the game.
Ban it: Probably the second-most common solution, though extremely prevalent in large games where the GM has too many players to keep track of to fine-tune them all. Simply banning certain character options makes it impossible to use them to unbalance the game. In 7th Sea, for example, we tried to ban all non-core book mechanics until the game had been up and running for several months (a ban that is still in effect on paper but has had so many exceptions made I call it a failed effort). The problem with this approach is that it upsets players who wanted to use those mechanics, and some players will always be tempted to try to have an exception made just because they want to be the exception; I'm guilty of this myself - after I got an exception made to play a Brotherhood monk in Heroes of Rokugan 3 I started plotting how to get the GMs to let me play a Heichi Bushi.
Ignore it: This is the easiest option though it can have the greatest consequences. The GMs say, "Yes, the game is unbalanced. Accept it and move on." The downside of this solution is that players can become upset if their supposedly bad-ass character is overshadowed by other characters due to unfair mechanics.
When it came to 7th Sea, I encouraged option 3. Even the core book is unbalanced so all banning supplements did was limit how many unbalanced options we'd need to deal with and how many new mechanics we'd need to learn. Furthermore, we were upfront with the players. This is a first edition, never updated, RPG written over a decade ago: gaming has made a slough of advancements since then, none of which had been incorporated into 7th Sea and we were not going to go back and re-write the system to provide balance. The system was there to allow us to tell a story, nothing more. If that would get in the problem of you're enjoying the game, this was probably not the right game for you. You want to know the most powerful options: here you go.
And this is the difference between unbalanced and broken: something that's broken makes the game less fun. Take the Minor Clan Schools from l5r mentioned above: while they are unbalanced (weaker) than the Great Clan Schools, they also allow one the opportunity to portray a very different character than normal in l5r. Now note that in this example the player is choosing to portray a character unbalanced to be less powerful than the standard counterparts. This is a key distinction; unbalanced options become broken when they allow one character to outperform the other characters, at which point the game becomes less fun for the players who are being overshadowed. It is the reaction of the other players that determines whether a character option is broken, not the mechanics of the option itself.
This is one of the reasons we did the post about why we would not be attempting to balance most of the 7th Sea mechanics: there would be too many. The work of balancing them would be arduous and keeping track nigh impossible; this is a new game for the GMs as well as the players and we can barely keep track of the real mechanics, let alone twenty pages of house rules. By warning people about this approach we were adjusting (or attempting to adjust) attitudes so that elements that might be considered broken were merely unbalanced.
Another aspect of the relationship between unbalanced and broken is how such things are introduced. If we accept brokenness as how game balance relates to fun rather than an extreme case of skewed balance, then we can take something unbalanced and keep it unbroken as long as it makes the game more fun. An example of this would be our introduction of a Crescent character. We out-and-out banned Crescent characters (and all associated mechanics) for the start of game but one player approached us asking for permission to play such a character. She did so with a backstory that justified her presence in the game and a concrete plan for how her character would interact with others in a way that would make the game more fun for everyone involved. Knowing the player and her gaming habits we trusted her and approved the character. In just one month this character has become one of the most heavily-involved characters in the game, creating an extraordinary amount of plot for herself, despite the fact that the GMs have had almost no opportunity to write plot to get her involved. Does she have access to character options other players don't? Absolutely. But that has enhanced the game for a dozen other players and given them more fun. Ergo, not broken.
In the end, it all comes down to what you consider the game to truly be. As I've said before, "anything that makes the game more fun is the right way to play the game, and anything that makes the game less fun is the wrong way to play." Game balance is simply one more tool in the arsenal of fun.