Inspired by some posts in this RPGnet thread, I got around to quantifying the most common definitions in the table-top community.
Much like Dungeons & Dragons alignment, the term "game balance" has many valid and popular definitions, but a lot of the time many folk do not have an agreed-upon consensus for discussion and end up misinterpreting the other side.
Class/Role-based Balance: The game has certain duties fulfilled by certain archetypes, where they are more or less experts in several skills to help supplement the whole group. Archetypes which can invalidate others by doing their job better, or ones which don't do their own roles very well, are imbalanced.
Mathematical Balance: One of the most common and well-known types, this looks at the underlying framework of dice-based probability and sees if there are any options which swing the variables too much or throw things out of whack.
Spotlight Balance: The idea that characters should have equal screen time in the gaming session, and that things which focus too much on one character is bad. Quite common in story-game RPGs.
Competitive Balance: Balance based on how a player's options relate to their ability to move against other players. Different from class and spotlight balance in that a lot of times individual character units are more likely to be able to "hold their own" due to less emphasis on cooperative aspect.
Versatility/Options Balance: Balance where the more open-ended an archetype's abilities are, the greater the impact they have in the game world. This quite common in rules-lite RPGs and 3.X D&D where you had a spell for every occasion.
Many RPGs incorporate more than one of the above options. Not all of these are mutually exclusive, and in some cases can fold into the others, such as class/role-based balance and versatility/options balance. The biggest melee damage bonus in the world won't save you if an opponent can outright nullify your attack by flying in the air, so measuring by one metric may not be all-inclusive.
Old-school Dungeons & Dragons relies heavily on class/role-based balance in comparison to its later incarnations. 3rd Edition Dungeons & Dragons and Pathfinder relies more heavily on versatility/options balance. 4th Edition Dungeons & Dragons had a big emphasis on mathematical balance. I cannot say for sure what metric's most common for 5th Edition, but now that there's an SRD I should find out soon.