I've decided to call the last dimension that I'm going to address in this column purpose. This refers to the reason the developers made the game in the first place, and what they hope to achieve with it. Purpose is just as independent of genre as theme, setting, or audience are.
One purpose is entertainment. Entertainment games are generally made for one or two reasons: to entertain the player and (in most cases) to make money.
Some (actually, a surprising number of) noble-spirited souls make games for entertainment without any hope of gain at all. A few misguided creatures also make games purely for money without regard for whether it entertains the player or not -- this purpose produces shovelware.
Serious games are, in Ben Sawyer's useful formulation, games to solve problems. This doesn't preclude entertainment; solving problems is simply an additional purpose of the game. There are many subcategories of serious game: games for education, healthcare, advertising, political propaganda, evangelism, and so on.
(A Christian-themed game that tries to persuade players to become Christians is also a serious game; but most Christian-themes games are entertainment for players who are already Christians.)
The designers of serious games occasionally have to compromise the quality of their entertainment in order to achieve their serious aim. For example, truly hardcore war games, as played by the military, aren't necessarily fun. But the better serious video games manage to accomplish their serious goal -- training, investigation, advertising, etc. -- and to be enjoyable as well.
Then there are games as works of art. Because art expresses the vision of an artist, a "game" (and yet again we see the limitations of that word) designed to be a work of art may, like a serious game, compromise entertainment for another goal.
Any video game must strike a balance between the designer's vision and the players' desires (see my column, "The Tao of Game Design" for more on that), and in my opinion any commercial designer needs to put the player first and himself second.
But art games, I recognize, are different. Again, it's a continuum, and the individual designer must choose for himself where he wants his game to fall along it.
These aren't the only purposes for which games can be made; they're just a few major ones. Super Columbine Massacre RPG is a satire; I don't know if it qualifies as an art game or not, but it obviously wasn't intended to make money or solve a problem, and perhaps not even to entertain. Satire may be another purpose in its own right.
So there you have a few of the myriad dimensions along which games may be measured: genre, setting, audience, theme, and purpose. I've chosen to address only those because those are the ones that resolve the problem of how to discuss games with Old West or contemporary urban settings, games for girls, Christian games, and serious games.
What's my authority for making all these pronouncements? Nothing, really, apart from common sense and 20 years of observation. If you're an old-timer like me, this all may seem obvious. But clearly it isn't, if some people are still struggling to understand it. I hope I've helped a little.