I think all humans play by intuition. We see the position on the board and intuitively know what type of move we need to play. Then we use calculating skills to narrow down our candidate moves, and some logic to decide which candidate move does the job the best.
If you mean intuition like Tal's sacrifices - brilliant though they were in practical games, I don't respect that style of play. It seems too much like guesswork.
I believe pattern recognition is intuitive, and that intuition is gained from experience. When applied to chess, most have to play the game often to recognize it. The prodigious recognize it in experience outside of chess, and apply it to the game they are playing. 
I think that in chess, intuition plays more of a role in helping us find which moves not to analyze: the moves that are clearly bad or merely inadequate. As a beginner, I was overwhelmed by the number of possible moves. It is an oft-cited fact that the human brain can hold between five and seven things at once in conscious thought... far fewer than the number of moves that could be made at any time. After years of play, my intuition filters out the majority of possible (but useless) moves and allows my focus my analysis on a manageable quantity of candidate moves.
That said, I think we have all been in the situation where intuition helps most: time trouble. Here again we can draw quickly on the memory of past games to say "this seems like a good move" when there is not enough time to analyze the corrent position. It's quite a thrilling thing, to do or simply to watch, when you sac a piece with 30 seconds on the clock simply because it looked right.
The more chess you play and the more positions you see and think about, the more you are able to apply ideas formed in other games to similar situations in the game you are currently playing.
People mock bullet chess and I accept it probably isn't the best training tool in the world, but I think it can be good for honing your chess intuition, simply because you don't have enough time to calculate anything concretely, and have to play only by instinct and pattern recognition.
Yes, intuition is founded in pattern recognition. You have a particular goal in mind, eg. breaking open your opponent's kingside pawn cover, or skewering his queen behind his king. You can't possibly calculate out all the possible responses, but having been in similar positions before, you know the type of moves that are likely to lead to the position you are looking for. Intuition also works in assessment. If you've drawn your opponent's king out into the centre of the board, there is bound to appear any number of combinational possibilities. The more you play, the more patterns you will recognize, and then you can rely more on your feel for these patterns to guide you. That is perhaps the real difference between amateurs and masters.
You know what to do since you have seen it or something similar before.
No magic about it.
I believe pattern recognition is intuitive, and that intuition is gained from experience. When applied to chess, most have to play the game often to recognize it. The prodigious recognize it in experience outside of chess, and apply it to the game they are playing.
Excellent reply. I mostly notice it when doing Tactics, though I'm not always right. What I hate, is getting the gut feeling that a particular move is right but not being able to calculate it correctly to see that it is the best move. Then I go and choose some other line and find the gut move was actually the right one.
Then there are those times when I go with my gut and I'm wrong 
I need to get a lot more patterns ingrained into my head and get a lot better at calculating to figure out when my intuition is actually right. Of course, I'm rated pretty low (considering) so what do I know 
The original poster seems to be equating aesthetically appealing moves with balanced positions, but I think the most aesthetically appealing moves are those that fly in the face of common sense. Being able to realize that the usual rules don't apply in a given situation, and seeking out surprising moves is an important skill, and is probably related to experience and thus intuition.
I agree... you never see Qxe3!! when it is an obvious move taking a hanging piece. It is only the ones that are difficult to find - usually because they appear so risky or unusual - that get the distinction of brilliant moves.
I wish I could respond with something better than this but;
All of your responses are well thought out and I would have to believe correct. The original post is a terrific one as it spurs one to consider. I do not think there are any incorrect responses to the original post. They all have merit and deserve pondering.
With that I will only thank you all. A special thank you to the original author of this thread.
You guys make chess live and breathe. I am not a strong chess player, just a patzer hopelessly in love with this game.
OK, time to answer to some answers:
rigamagician - post #10 - when I said "balanced", I also said 'visually' with it. But nevertheless, let me explain: I was referring to positions that are not by any means necessarily stable, but that are essentially homogenic when you look at them. That means the pieces of one army in, say, a rich middlegame, should be harmonious (well-connected (without any pieces too far away), supporting each other as much as they can without suffocating, with activity and in appealing formations (Knights defending each other, batteries etc.)), and the pawn structures should have mostly connected pawns, healthy chains, and should support the piece formations well. Yes, those all sound like characteristics of positions that are both balanced AND stable, but many stable positions are appealing AND drawish, if not drawn. Instead, I'm talking about positions which have got imbalances, but are still pleasant to watch, so the right moves can come more intuitively. For example, the main lines of both the KID Classical and the Sicilian Sveshnikov are both quite sharp and full of imbalances, but the wing avalanches of whole armies resembling yin-yang around the closed center in the KID appeal to me much more than those scattered-looking positions of the Sveshnikov, with many holes, doubled pawns, exposed castles, and hardly any central symmetry. Of course, the sheer beauty of the logic should guide us through both openings, but since the KID is prettier, it should have more support by our aesthetical intuition, and you've got ugly moves even there! Of course, it's also essential that possibly ugly, but unexpected and brilliant moves exist in pretty positions, so they can contrast the apparent peace, break the monotony and in the end make that game and the opening more beautiful!
FredtheCat - you're very welcome, and we most sincerely thank you for your compliments to the thread!
El_Senior - you're right, there is also such thing as negative intuition! 
I think George Lucas elaborated on this one. 
I believe pattern recognition is intuitive, and that intuition is gained from experience. When applied to chess, most have to play the game often to recognize it. The prodigious recognize it in experience outside of chess, and apply it to the game they are playing.
This is almost exactly my instructor's view. He advises hours of tactics training and playing through master games to develop it. He practices what he preaches: 400 tactics puzzles alone a day
Is it possible to play beautiful (surprising, sharp) moves in ugly positions (with doubled or backwards pawns)? I would say yes, and I possibly appreciate the beauty of such moves all the more.
To answer the question, is it possible to play intuitive chess without deep knowledge of chess? No.
See some of the above discussions on how intuition is formed, and you will understand why.
One cannot have intuition about something without being intimately familiar with the subject.
This is just wrong. Intuition has a bit of a mystery about it simply because some people know a subject without having even been exposed to it before. They seem to be able to apply their experience in other areas to an entirely new subject, and immediately grasp it.
Well, you can have intuitions about a subject that you don't know well, but they will likely be incorrect intuitions.
I am guessing that intuition will have to do with (partly subconscious) transfer of knowledge from one area to another. For example, you intuitively feel that a certain attacking plan should work, and without calculating all the lines, because it worked in similar (though not exactly the same) positions. Such an intuition could prove correct or incorrect.
I guess I'm going to have to cite my sources.
Starting with Erik Kandel, who showed how memory works in In Search of Memory, and moving through such a variety of sources as Robert Hogarth's Educating Intuition (available online through this link: http://books.google.com/books?hl=en&lr=&id=fsffJGkpwVIC&oi=fnd&pg=PR7&dq=intuition&ots=_bV-NDk80b&sig=rbRLrfi6e9pqaI50tqn0SVsIi5o#v=onepage&q&f=false )
I don't know how you would explain the child prodigies obtaining "a wealth of knowledge" about chess. And the word "intuition", has a philosophical, and psychological history, which precludes me from citing sources. I thought the question was to us, not Krogius?
One can know nothing about a subject, yet by reason, based on experience, can intuitively "know" something is lacking. For instance, when inventions occur, or when modifications are made to things, sometimes those inventions occur to one who has very little knowledge of the subject, and it's history.
Intuition is a form of knowledge, not a substitute for knowledge.
What does that mean? One may intuitively understand other subjects that are new to them. So even if one is not familiar with chess, pattern recognition can occur in other areas, and be applied to the game. That is why one may be quite prodigious at chess, w/o having very much experience with it.
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A question for all of you players who have experienced intuition in chess, applied it in one or more chess games, and know it. Maybe I've had my intuitive moments while playing chess, but I don't recognize them. I'm also asking, because one of the members of Chess.com said this in his blog: "Can one play intuitive Chess without deep knowledge of Chess itself? Well, to a certain extent, the answer is yes. If one, for example, understands war strategy, he/she can use analogy and guess good moves without a deep knowledge of the game.", and I was wondering if tending towards playing aesthetically appealing and visually "balanced" positions is a good analogy, since in chess with all its depth and beauty, many of the best moves are often ugly and thus harder for me to find, since they demand seeing logical beauty before the spatial visual one (if there is such in the position)! And logic only demands (life) intuition in the form of logical intelligence, but completely transferring that intelligence to a visual dimension isn't easy at all! In fact, I don't quite know of any person that can rightfully say to have ever managed to achieve that! Thus, I think that, if you're not well-used to pretty AND ugly positions, like high-level pros (who've got it all figured out both by logic and some intuition), then the visually aesthetical side of intuition can really be wrong in its incompleteness. What do you think? And how should intuition feel anyway?
Edit: This thread has become a bit long now, so if you don't want to read it entirely, just read some of the first posts by users Fezzik, trysts and Atos, and it should give you a nearly full picture about the main debate throughout this thread. Instead, however, I'd like you to focus more on the second part of this thread's title, because just about anything additional on the here already discussed topic is going to be repetitive anyway.