The Power Gamer.
The Power Gamer wants to make his character bigger, tougher, buffer, and richer. However success is defined by the rules system you're using, this player wants more of it. He tends to see his PC as an abstraction, as a collection of super powers optimized for the acquisition of still more super powers. He pays close attention to the rules, with a special eye to finding quirks and breakpoints he can exploit to get large benefits at comparatively low costs. He wants you to put the "game" back in the term "roleplaying game", and to give him good opportunities to add shiny new abilities to his character sheet.
The Butt-Kicker wants to let off steam with a little old-fashioned vicarious mayhem. He picks a simple, combat-ready character, whether or not that is the best route to power and success in the system. After a long day in the office or classroom, he wants his character to clobber foes and once more prove his superiority over all who would challenge him. He may care enough about the rules to make his PC an optimal engine of destruction, or may be indifferent to them, so long as he gets to hit things. He expects you to provide his character plenty of chances to engage in the aforementioned clobbering and superiority.
The Tactician is probably a military buff, who wants chances to think his way through complex, realistic problems, usually those of the battlefield. He wants the rules, and your interpretation of them, to jibe with reality as he knows it, or at least to portray an internally consistent, logical world in which the quality of his choices is the biggest determining factor in his success or failure. He may view issues of characterization as a distraction. He becomes annoyed when other players do things which fit their PCs' personalities, but are tactically unsound. To satisfy him, you must provide challenging yet logical obstacles for his character to overcome.
The Specialist favors a particular character type, which he plays in every campaign and in every setting. The most common sub-type of specialist is the player who wants to be a ninja every time. Other specialists may favor knights, cat-people, mischief-makers, flying characters, or wistful druid maidens who spend a lot of time hanging about sylvan glades with faeries and unicorns. The specialist wants the rules to support his favored character type, but is otherwise indifferent to them. To make a specialist happy, you have to create scenes in which his character can do the cool things for which the archetype is known.
The Method Actor
The Method Actor believes that roleplaying is a medium for personal expression, strongly identifying with the character he plays. He may believe that it's creatively important to establish a radically different character each time out. The method actor bases his decisions on his understanding of his character's psychology, and may become obstructive if other group members expect him to contradict it for rules reasons, or in pursuit of a broader goal. He may view rules as, at best, a necessary evil, preferring sessions in which the dice never come out of their bags. Situations that test or deepen his personality traits are your key to entertaining the method actor.
The Storyteller, like the method actor, is more inclined to the roleplaying side of the equation and less interested in numbers and experience points. On the other hand, he's more interested in taking part in a fun narrative that feels like a book or a movie than in strict identification with his character. He's quick to compromise if it moves the story forward, and may get bored when the game slows down for a long planning session. You can please him by introducing and developing plot threads, and by keeping the action moving, as would any skilled novelist or film director.
The Casual Gamer
The Casual Gamer is often forgotten in discussions of this sort, but almost every group has one. Casual gamers tend to be low key folks who are uncomfortable taking center stage even in a small group. Often, they're present to hang out with the group, and game just because it happens to be the activity everyone else has chosen. Though they're elusive creatures, casual gamers can be vitally important to a gaming group's survival. They fill out the ranks, which is especially important in games that spread vital PC abilities across a wide number of character types or classes. Especially if they're present mostly for social reasons, they may fill an important role in the group's interpersonal dynamic. Often they're the mellow, moderating types who keep the more assertive personalities from each other's throats -- in or out of character. I mention the casual player because the thing he most fervently wants is to remain in the background. He doesn't wnat to have to learn rules or come up with a plot hook for his character or engage in detailed planning. You may think it's a bad thing that he sits there for much of the session thumbing through your latest purchases from the comic book store, but hey, that's what he wants. The last thing you want to do is to force him into a greater degree of participation than he's comfortable with. (Of course, if everybody in the group is sitting there reading your comic books, you've definitely got a problem...)