Types of Games
DOOM-like first-person games— These games are full 3D, for the most part, and you view them from the character's perspective. DOOM, Hexen, Quake, Unreal, Duke Nukem 3D, and Dark Forces are all good examples of this type of game. Technologically, they're probably the most difficult to develop, and they require cutting-edge technology to be noticed.
Sports games— Sports games can be either 2D or 3D, but these days more and more are 3D. In any case, the sport can be one-man or team play. The graphics in sports games have come a long way. Maybe they're not as impressive as first-person games, but they're catching up. However, the artificial intelligence in sports games is some of the most advanced of all the game genres.
Fighting games— Fighting games are typically played by one or two players, and the action is viewed from the side or by a floating 3D camera. The game imagery can be 2D, 2.5D (multiple 2D bitmap images of 3D objects), or full 3D. Tekken for the Sony Playstation I is the game that really made the genre for the home console market. Fighting games aren't as popular on the PC, probably due to the interface problems with controllers and the two-player fun factor.
Arcade/shoot'em-up/platform— These games are your typical Asteroids, Pac Man, and Jazz Jackrabbit type stuff. They're basically old-school games that are primarily 2D, but they're being redefined and remade into 3D worlds. However, the gameplay is relatively the same as it was in 2D.
Mechanical simulations— These games encompass any kind of driving, flying, boating, racing, and tank-battle simulation, and any other kind that you can think of. For the most part, they are 3D and have always been (even though they didn't look good until recently).
Ecosystem simulations— These games are really new and have no real-world analogs—other than the real world itself. Here I'm talking about Populous, SimCity, SimAnt, and so on. These games allow you, the player, to be a god of sorts and control an artificial system of some kind, whether it's a city, a colony of ants, or a financial simulation like Gazzillonaire (very cool game, BTW).
Strategy or war games— These games have splintered into a number of subgenres. But I'm not religious about them, so suffice it to say we're talking about strategy, turn-based (sometimes), thinking types of games such as Warcraft, Diablo, Final Fantasy, and so on. Again, I'm being a little cavalier here since Diablo is real-time, but it still involves a great deal of strategy and thinking. On the other hand, Final Fantasy is turn-based and not real-time.
Interactive stories— This category includes Myst-like games. Basically, these games are prerendered or on "tracks," and you move through them by figuring out puzzles. Usually, these games don't allow free roaming and are like playing interactive books, for lack of a better definition. Moreover, they aren't really "to-the-metal" game programs because 99% of them are written using Director or a Director-like tool. Boring, Jules.
Retro games— This area of gaming has sprung up overnight. In a nutshell, there are people who want to play old games, but with more bells and whistles than the originals. For example, Atari has made about 1,000 versions of Tempest. Granted, they never sell, but you get the point. However, I have had a lot of luck remaking some of the old games like Dig Dug, Centipede, Frogger, and so on.