Definition: What is Depth? A multiplayer game is deep if it is still strategically interesting to play after expert players have studied and practiced it for years, decades, or centuries.
We often talk about skillcaps. The more skillcaps a game has the better it is. Ideally, you want a game to have an infinite number of skillcaps. A skillcap is a level of expertise which allows the players in it to dramatically beat the players which belong to the lower level of expertise. The idea behind skillcaps is that, if you, as a player, are in a lower skillcap, it means you're not entirely exploiting all the possibilities of the game. Many skillcaps means that increasing things like reflex is not enough. The player needs to increase its understanding of the game. Incidentally, that game is very deep.
Also, there is a measure for player expertise called the Elo rating system: http://en.wikipedia.org/wiki/Elo_rating_system
Definition: What Is Balance? A multiplayer game is balanced if a reasonably large number of options available to the player are viable--especially, but not limited to, during highlevel play by expert players
Well, not really in my opinion. A game is balanced if you cannot find a single way-of-playing to beat an equally-expert opponent more than 50% of the time when the opponent is not allowed to use that same way-of-playing. (Note we are talking here about robots which automatically play each game with the exact same level of expertise.)
EDIT: oh the author calls that fairness. Fair enough then.