Thinking more on this... Gaming genres are an artificial construct, so there is no natural way to categorize games, and there will be much cross-pollinating. Elements from a popular game will often be borrowed by a number of others, even if they have a completely different genre. If one were to actively categorize games by "genre" you would need to identify a number of variables. Here's the basic model I suggest:
1. Presentation: How do gamers see the game? There are two subsets to this:
a) Dimension: 2D, 3D or the always myserious 2.5D (4D, despite promises to the contrary, has yet to appear. No one seems interested in 1D).
b) Perspective: Single screen, scrolling screen, first person, third person, fixed camera, free camera, on rails, top down, etc.
2. Setting: What is the game world? (as with Perspective, these can be easily be blended ie - SF Horror)
a) None/irrelevant: Puzzle games
b) Contemporary: Set in a world somewhat modeled on the one we know and live in, give or take about 20 years.
c) Historical: Set at some point in the known past (Old West, WWI, Ancient Rome, etc)
d) Science Fiction: Ray guns, aliens, and psychic powers
e) Fantasy: Magic, swords, orcs and dragons
f) Horror: Werewolves and Vampires and Zombies Oh MY!
g) Cartoon/Classic Video Game: Mario, Sonic, Jak and Dexter
3. Gameplay: What are the gamers doing? This is what most of us think of as the game's genre. Games may borrow some gameplay elements from other genres (Sports sim, Action RPG, etc)
a) Sports: A simulation of a sports game, usually based in the real world.
b) Racing: A driving game. Typically broken down into Arcade and Sim
c) Adventure - Puts more emphasis on exploration and story telling than action (Zelda).
d) Action: Meant to keep the adrenaline running. Ninja Gaiden, God of War, etc
e) Shooter: A subset of action that has grown large enough to deserve its own entry. Many games have shooting elements, but true shooters put the emphasis on... shooting.
f) Stealth: A mutant offshoot of the action/shooter, which has replaced much (but not all) the action with sneaky elements
g) Sim: Simulates a real world activity (flying a plane, living life, running an amusement part, dating)
h) Strategy: Often a complex derivation of Chess and Risk. Main goal is development with an eye towards conquest.
i) Music/Rhythm j) RPGs - Player grows, develops and customizes a character or group of characters (most often against an epic fantasy setting).
k) Minigame compilations - Includes many short, often unrelated games with (often) minimal depth (if Multiplayer, called party game).
l) Puzzle - Emphasis on problem solving with minimal or absent characters, action and story.
m) Platformer - Emphasis on moving from one disconnected platform to another in order to travel through the game world. Three common subsets would be the puzzle platformer (Braid), skill platformer (Mario), and action (adventure?) platformer (Prince of Persia).
n) Survival - Typically a horror title (anyone have an exception?) in which ephasis is placed on survival rather than defeating enemies.
o) Point-and-click - a blend of puzzle and adventure game.
p) Exercise games - Thanks Nintendo...
q) Text based: Nearly extinct form of gaming from paleolithic era. Pioneered 0D gaming.
4. Community: Who plays the game? 1 player, 2 player, online, co-op, etc
I think between these four factors, you can pretty much describe any game.
