Concept
The premise that Richard Bartle laid down in 1996 that players can be categorized into 4 different primary groups is the basis of this particular taxonomy. They are as follows: Achievers, Explorers, Socializers, and Killers.
There is an 8-dimensional version which sub-categorizes the players types as being either explicit and implicit.
Achievers
Achievers are the types most likely to engage in "grinding" or "farming" behavior. As the title implies, their primary source of satisfaction or reward comes in the form of attaining "achievements", which can come in any form. This translates into rankings as well, though not necessarily PvP rankings, especially if their Killer quotient is low. However, those with both a high achiever and killer set of quotients tend to be drawn to PvP MUDs or spend much of their times in PvP play.
This is supposedly one of the most common player types, with players concentrating on gameplay that involves gaining levels, amassing wealth, having the "best" gear, and even being the first to either solve something, attain something, or beat something.
While this tends to be a very solo-centric style of play (with exceptions such as PvP), the multi-player component is important because it gives the player the chance to show off their achievements. This is a very integral and important part of the process that sometimes does not translate to non-MUD games. (An achiever type in a MUD may not manifest the same type when playing in other types of single-player games.
Explorers
Explorers tend to care about the world-building, backstory, storylines, and lore of the game. As the title implies, their primary source of satisfaction or reward comes discovering things for themselves.
This is also the player type most commonly associated with finding cheats, loopholes, bugs, and un-intended features. They may or not not make use of them, and depending on their Socializer quotient, may share those secrets. A high achiever may share them also, but the reasoning behind it would be to show off.
Explorers with high quotients also tend to immediately "move on" or drop a game once they have "beaten" it, or "seen everything". Alternately, Explorers that see a game from a different perspective (such as the meta of a game) may spend far more time, even years, playing a game that will have exhausted all other player types.
Killers
Killers are your PK types. They simply get more out of competing with real players than they do against AI, no matter how intelligent. The Killer type is however, not necessarily evil. Though your griefers, gankers, and pwners fall into this category, there are also those who are capable of being civil, fair, and honorable. Like the Richard Connell's The Most Dangerous Game short story from 1924, the thrill of the hunt is no there without the human element. Only the greatest hunter can be the greatest prey, and that is Man.
Socializers
Socializers are the most common type (unsurprisingly) on the social MUDs, the MUSHes, the MUSEs, the MOOs, and Tinys. Their primary source of reward from playing comes from interaction with other players. However, they are very well represented on even the most hack-and-slash or combat-oriented games. They are also attracted to the larger, more popular games because they may score higher in their other quotients and see other interesting components. Largely, however, they will at least be attracted because of the high concentration of players, thus providing many opportunities to join or begin a community.