"Twitch" is a generic term representing the importance of manual reflexes to play a particular game. An example of twitch is aiming your gun in a first person shooter to hit a target. Another might be timing a jump to get onto a moving platform in a Mario type console game. Still another example would be the ability to quickly and efficiently coordinate troop movements and orders in a micromanaged Starcraft battle. All twitch play involves hand-eye coordination, fast decision making, interface control, and after a while, automatic reflexes burned in by hours and days of play.
Many games rely on twitch to play well. First person shooters are primarily, if not completely, twitch games. Almost all of the early arcade and console games (Pac Man, Defender, Galaga, Sinistar, etc.), not to mention pinball, are twitch based. As games evolved and a larger, maybe older, market was sought, twitch became less of a factor, or a total non-factor. Games where twicth doesn't matter are typically turn-based games like Solitaire, Myst, turn-based strategy, etc.
The modern computer-based trend is generally a combination of the two. The games that are turn based are obviously complete non-twitch games. However, turn-based play is impossible in a modern MMORPG (the Realm's combat system notwithstanding: it is really a graphically-enhanced MUD). It's only barely tolerable to wait for others to take actions in a turn-based game involving as few as 3 people; in a playing arena with 2000 simultaneous users, it would be insane. It would stand to reason, then, that in such a realtime environment twitch would be an important part of player skill. However, in MMORPG design and player attitude, there is a decided distaste for twitch play, and this is reflected in the game engine.
The distaste for twitch play may stem from a prejudice on the part of RPG nazis who feel that the CRPG is somehow above the FPS, and thus should be above the FPS's reliance on twitch. There is some justification for lowering the importance of twitch in a progressive game that incorporates character building; after all, twitch depends on the abilities of the player, not the experience and design of the character. Nobody wants their level 10 character trounced by a level 1 character whose player can click the mouse faster. As a result, the importance of twitch has been vastly reduced (though not completely eliminated) in the play of the modern MMORPG, and as games "evolve," new mechanisms may be introduced to further lower the impact of good twitch play, such as UO's inclusion of Last Target into the client, eliminating the need for players to manually select a fast-moving target for spellcasting (or use a third party program to do it for them). The need for Last Target in UO was somewhat related to the awkwardness of its interface, however, and cannot be dismissed entirely as a cheap way of "making the game easy."
However, trying to do away with the importance of twitchy character control is a mistake when carried too far. In the early days of Asheron's Call (say up through about month 6 or so of final), character control was an important part of being a good player. Characters who were weaker "on paper" than the guy next to them could do extremely well in difficult situations if their control was good. High-powered characters whose players just let them sit still and auto-attack were more likely to meet an ignominious death. Because Asheron's Call has balance issues and no cap on player power potential, the importance of twitch went away. No amount of eye-hand coordination can compensate for the levels of raw power and invulnerability that players and monsters alike achieved. The long-term success of a character today in AC is determined almost entirely on the character's initial design and how much experience he has. The role of character control is limited to only the closest contests of power, maybe within a slightly wider range in the realm of PvP, but twinking and a good macro count for far more now than a player's actual ability to react to his environment.
Increasing the importance of twitch play in the MMORPG allows some reward for the player who is actually good at playing the game in this manner, and helps to differentiate otherwise cookie-cutter clones from each other. This is not to say that twitch should be overwhelmingly important next to character development: nobody expects a total novice character to be able to whip an enemy way over his head just because he's got good reaction time and makes quick decisions. However, in combination with limiting player power, it can make all the difference for a truly excellent player at the top of the game's power curve, as opposed to someone who got there without mastering the interface.
Twitch also helps a game's longevity. As long as twicth and automatic reaction time are important, there is always room for improvement as a player. One can powerlevel forever and read endless messae board posts about what theoretical combinations and tactics work well, but if the ability to execute these tactics is a function of the player's ability, there is always something to work for.
This is my opinion, not much as a player but more as a Game designer. I know that it can be relative to each player experience so I think it's a good discussion.