black is giving up his bishop pair to double white's pans on the c-pawn, I'd have probably done the same thing, it weakens both the pawns. You usually want to avoid doubled-pawns whenever you can.
Undefineable Advantage

A good starting place: http://en.wikipedia.org/wiki/Nimzo-Indian_Defence
Basically, black's idea is to take a lead in development and/or double white's pawns in exchange for the dark-squared bishop. Black will then try to create a position that favors the knights and/or win back a bishop from white.
Take what I say with a grain or two of salt. For some reason, even though I like the idea of the Nimzo-Indian, I don't play it very well.

Cool, thanks for your comments folks. I guess the problem I have is that I can't see which of the concessions is worse - the doubled pawns or the loss of the bishop pair.

@ Estragon - Well, i've started playing it as white, merely as an experiment. Perhaps unsurprisingly, I've had some embaressing losses! I think my problem is that I have enormous difficulty forming plans and thinking strategically, which sort of renders any opening irrelevant. I think as white, I've tried to use my bishops too aggressively, and ended up not being very well developed. I gues I have a lot of work to do!

Is the structure weak or does it lack mobility? Or does lack of mobility make the structure weak?
Doubled pawns should not be "avoided", many times they are even beneficial as they can give you files, the center and minority attacks. Take a look at the Sicilian Shveshnikov(?) If you want a good example. This is the problem with alot of begining players they memorize positional "advatages" like doubled pawns and the bishop pair and then apply it out of context. If you want to get your hands wet with positional knowledge take a look at My System. Good book.
Also I beleive 4.a3 in Nimzo is proven to be advantagous to black.

There was some glitch in the forum page with the game I posted, so I'm putting additional comments here.
My final note in the game, reads "Sweet! Black cannot defend c1=Q and defend the rook on f3 both. 1-0!" Obviously c1 should be c8 and f3 should be f6, I flipped the board in my head for some reason.
Errant. Keep reading Silman. I very much agree with Chess enigma's comments about doubled pawns and applying advantages "out of context" as he puts it.
Positional advantages/imbalances won't win the game FOR you. You must use them! And if you can't find a way to use them, they aren't an advantage. (Likewise, a weakness your opponent can't/doesn't exploit is no weakness at all)
Nimzo Indian is a beautiful opening with many beautiful games. I think it's the Ruy Lopez of 1.d4 -- It's deep and complicated and in general, the better player wins. There is no undefineable advantage to it, for either side! There is no simplistic answer after four or five moves in the Nimzo to who stands better. It's a perfect opening to learn as you read Silman because Nimzo leads to asymmetrical, imbalanced, positions. The player who better knows how to maximize his advantages and exploit his opponents disadvantages is the player who wins.
I've just started to read 'How to reassess your chess', which starts with a scathing attack on the reader as to why he or she is still rubbish at chess. Since I fall in to this catagory (I am a far weaker player than I should be), I have organised my weakness in to two parts. Firstly, I have almost no strategic awareness. Secondly, I do not understand opening theory. Specifically, I would greatly appreciate advice on the following: