I moved Tamsin's remarks and the discussion that followed from playing styles and player types, which is a joke (so is this page, but that is not it's intention). I hope I did not offend anyone by moving their comments.
Tamsin takes a slightly more serious look at playing styles.
I think people tend to gravitate toward one of two bases: they either prefer profit (the territorial style] or they prefer influence, thickness and power, from which they aim to earn points in the course of middlegame fighting and the endgame.
There are various degrees to which these two fundamental styles can be taken. At one extreme, Kobayashi Koichi likes solid territory and his fuseki and middlegame decisions emphasise this very strongly, to the extent that his Go can appear somewhat "stingy" and unspectacular (it's certainly effective, though, as his impressive list of titles proves). Takemiya Masaki, represents the other extreme. He once famously derided Kobayashi's style as "subway Go", and created the beautiful, but inimitable, "cosmic style" in which he generates huge moyos in the centre, while taking a somewhat relaxed attitude to his opponent's profit. This style has also proved itself extremely effective: Takemiya has not only earned the adulaltion of Go fans across the world, but even more importantly, he has won many titles. Perhaps, occupying positions closer to the middle of the spectrum, are players like Otake Hideo, who stresses thickness in the opening and uses it gradually to overhaul his opponent's lead and Yi Chang Ho, who often takes early profit, but is not, repeat not, an easy customer to deal with in a fight!
There are different interpretations, if you like, of the two styles, which become new styles in their own right. Sakata Eio and, later, Cho Chikun, liked to grab territory, while leaving groups exposed to attack, relying on superb reading and defensive skills to save the weak groups while maintaining sufficient territory to win shinogi. This kind of Go might be dubbed the "greedy style". Another distinctive style is that of O Meien, whose Go is characterised by imaginative openings in which speed is of the essence, original shapes (Meienisms) and heavy-duty fighting skill.
So, what about your style? As I stated above, people gravitate either to thickness or to territory. How much of this do you think is personal choice and how much is unconscious bias? (Please discuss below.) My feeling is that below pro level, one has only limited choice in one's style: I have found that I win most often when I stress influence over early profit, and that even when I play moves especially designed to take territory, I somehow end up producing central moyos -- I have won a surprising number of games in which I played two 3-3 points only to end up building large-scale territory in the centre. Do you think it is wise to copy your favourite pro's moves? For myself, I would like to make Meienisms, but suspect that I would end up only creating difficulties for myself because I'm not O Meien the 9 dan professional go player, after all, but Tamsin Jones the musicologist :-)
I look forward to reading people's responses and thoughts upon this fascinating area of go appreciation!
Alex Weldon: I disagree that it's impossible to have one's own style below pro level, although perhaps at weaker levels, it's more a matter of what you're good at and what you don't understand, rather than a choice of style. On my personal page, I describe my style a bit; in a nutshell, I like to play the Manchurian fuseki, build big side territories and moyos, then invite invasion to build strength and use it to wipe out any territory my opponent has made.
I definitely play differently than most players my level (6k*) on IGS, and there are several people I know by name (whether they remember me or not, I don't know) because they have their own styles. "Takakiyo," for instance, likes to cut the board up into lots of small groups of his and his opponent's, because he's very good at life and death; he rarely takes much territory, but instead relies on his ability to eventually kill something. "Yaoyu," on the other hand, likes to play second line "submarine" moves inside your prospective territory, letting you build a central moyo (which he'll later invade) and denying you the opportunity to build anything at all along the edge.
Tamsin: Sorry, Alex, I did not mean that one cannot have a style of one's own, but rather that I suspect that one has little choice in what that style will be. So, for example, a fundamentally "territory" person will unconsciously be biased toward profit even when he uses influence-based moves in the fuseki. Your friend "Yaoyu", for instance, will probably remain aggressively territorial and inclined to invade early even if he gets a lot stronger, although the way this shows itself might become somewhat sublter.
Alex Weldon: True, but do you really think the pros "choose" their style? I suspect that a style is always developed naturally, rather than chosen. I seriously doubt that, say, Takemiya Masaki could overnight decide to play territorially and be as good at it as his peers.
I imagine that whatever style you start to develop during your days as a kyu player will be gradually refined and improved upon until you are as good as you'll get, and this is probably true in the case of the pros, too. If they played territorially during their days as a student, they'll probably play that way as a pro as well.
It would, however, be very interesting to see a friendly game between say, Takemiya and Kobayashi, where they agreed beforehand to try their best to play in the other person's style, just to see how flexible pros really are. :)
Tamsin: On the contrary, pros are able to choose and change their styles. What I said above about style being the result of unconscious biases applies much more to amateurs. Kobayashi Koichi was a very aggressive attacker early in his career, but achieved his greatest successes after he adopted his no-nonsense "subway style" of territorial Go. Similarly, "Killer" Kato Masao became "Endgame" Kato later in his career. Finally, look at Kitani Minoru as an example of a player whose Go changed dramatically over the decades.
I do agree with your other comment: but I can't imagine Takemiya being able to bring himself to play like Kobayashi. Besides, as far as I'm aware, they're not the best of buddies, so that kind of friendly match would never happen, sadly.
Benni: I did not think that i have a style. Instead i am stil experimenting with sometimes a more territorial approach and sometimes a more moyo-like style. I have lost many of my games because i played to territorially and many because i played to moyo-based. Than in the next game it is the other way around. Often i begin with territory oriented moves in the Fuseki and finally end with a large moyo, which i am unable to defend and vice versa. Maybe one day i found the right Balance.
Patrick Taylor: I think it might be a fun and interesting idea to create a kind of Styles of the Deshis? page. The idea would be to set up a few basic board positions from joban or chuban and see how people respond to the position without any outside influence. I don't think I'm necessarily clever enough to come up with the positions, since they would have to have a great deal of flexibility, but I'm sure someone here who thinks this is worthwhile could rise to the challenge. The positions themselves would have to allow numerous responses. The real hardship on the part of the participants would be withholding critical commentary of others' choices.
The 'style demo' should consist of three parts: fuseki preferences, late joban development, and a chuban example that showcases the contributor's large-scale attacking methods. Each person should provide commentary on why they chose the moves they made. Of course, this page would be entirely for fun, since any kind of analysis of the effectiveness of the moves offered would defeat the purpose of such a style showcase.
TenukiSword WHen I play ON KGS, I tend to tell people what their style is based on the 12 Chinese Zodiac Animals. I still haven't worked on the concept, but I have a pretty good idea for it. Stay Tuned! The Styles of the Zodiac
 In the sense of this remark by Charles: When he (O Meien) was in a match against Cho Sonjin, it was said that the public didn't take to the pairing, for reasons of style (if you believe that). They're both 'messy' players, but O is seemingly more original.
 To start this I just compiled a list of aspects and style from professional players go styles. Actually I was looking for a list such as this one here when I found that one. As I am a beginner DDK myself, I would very much appreciate descriptions, the removal of doubles, addition of nuances and other modifications by more advanced deshi.