I am torn between the idea that dancing "teaches" a skill, and that dancing *is* a skill that give the chance for certain characteristic traits that are needed for life (what we can call "life skills") to appear and be developed further. DrDoug and fascination are both correct (as are all the posters who have replied before me) in naming and listing these certain traits. Dancing definitely provides the opportunity for all of these to be well developed. But to say that it definitely teaches them? I suppose it depends on how we use the words "teaches" and "cultivates," and if we can use them interchangeably, as it were.
I argue that we can, as to me, when something is taught it is exactly that: the possibility of an innate trait or skill to be developed further and emphasized. For example, as a Language Arts teacher, when I teach critical thinking skills all I can truly do is present the student with the text, the skill of inferring, and a chance to do just that. I did not teach them that skill, then. All I did was encourage them to sharpen and utilize more a skill that was already present.
Using the definitions as interchangeable, then, I would argue that dancing teaches the most important of all life skills: understanding (I will admit that patience, and the [affirmation of the] desire to get better are close seconds, and are tied into understanding). All of the traits that fascination mentioned are definitely needed in dancing, and dancing is a skill that forces the student to depend on those more than any other(s). But without patience all of those are useless. Without the desire to improve, even patience is useless in a sense. To become a better dancer one must ultimately learn to have a great deal of patience with oneself, one's partner, one's teacher... and by extension the competition organizers, other couples, judges, the DJs... and by extension all other things that come with our sport, including eventual travel, traffic, etc.
Patience is the one skill that makes life not merely bearable but livable and dare we say enjoyable. What was it that Nietzsche said once, "He who has a why to live can bear almost any how"? Well, the why of the dancer is always improvement. The how is patience. And undoubtedly, as I have said, this is the skill that is beaten into a dancer from day one, whether it is through constant comparison to others, to self, or to coach, or to whom and what-ever.
As one learns how to dance one learns to understand, pure and simple. One learns to understand that patience is required. One learns to understand that it is all an illusion. One learns to understand the self, and how s/he works, and then how to be sensitive, communicative, responsive, humble, etc....
Without understanding life is quite challenging. Whatever your understanding comes to be, and however it comes to be, it cannot be disputed that if one is a dancer, the sense of understanding (and by extension, appreciation, and patience) is, in a sense, far greater [than that of, say, a non-dancer].
This goes hand in hand with a theory I have proposed in past posts, that dancing is a proverbial ego-check, and a continual journey into, and the continual discovery of, the self.