I'm playing black. According to the computer analysis, the score is over 3 at around move 20 - and (piecewise) we're even! 

Any insight into where exactly I went wrong would be helpful.

The game:

http://www.chess.com/livechess/game.html?id=303414319 

42... Qg4+ ;  followed by Rh3mate.

White was looking good until move 39, Re7,  when he failed to deal with the threat to the h3 pawn...then you had it won until missing the move stated by MorganGray above, 42...Qg4+

41...Qxh3+?? Much better is 41...Rxh3+ 42.Kg2 or Kg1 Qxg4#.  Ya shoulda mated him! 

Hardly relevant to where he went wrong!

41...Qxh3+ only mates one move sooner than the move he made.

You consider missing a mate in 2 hardly relevant?  Maybe I have been playing this game for all the wrong reasons.

check2008, This is no reflection on you.  You still have alot of work to do.

If I can suggest a good place to start is the basic checkmate endgames (K+Q v K, K+R v K, K+2Bs v K, K+B+N v K).  You need to practice these until you can do them in your sleep.  It will take about 3 months.  

I call the engame technique in the basic checkmate endgames the "corralling method."  It is essentially the same technique in all of the basic endgame checkmates.  The process consists of using the power of the piece (ex. the R, no matter where it is on the board from the square it is on, it controls all of the squares on the file and rank) to form a fenced in area inside of which is the enemy King and canot escape. The next step is to make the fenced in area progressively smaller with the help of one's K and thereby drive the enemy K into the corner where you deliver mate.

I can garuantee you, if you do what I suggested, the next time there is a mating net in a game you're playing, like the one you missed in this game, it will jump up off the board and smack you on the forehead in a flash.

Good Luck/Hard Work in becoming a 'professional gunslinger' (a very strong player) if that is what you really want.  

The OP asked "Where'd I go wrong?"...In case you genuinely don't understand, here are the facts:-

After White's move 42, it was mate in 2 for Black. He played 42...Rg3+, instead of Qg4+ and the game was then lost.

From the game generally, I'd imagine he knows the basic checkmates.

What is important is not whether he knows the basic checkmates.  He may in fact know the basic checkmates.

It is a matter of visualization.  Being able to do the basic checkmate endgames in your sleep builds visualization patterns in your brain, just like tactics training does.  It is those visualization patterns that make the mating net in the game you are presently playing,  jump up off the board and smack you on the forehead in a flash.

Or it may be that this game, he had a rare oversight.  Only check2008 can tell us that.

Well, I was really referring to where I went wrong before move 25 or so. It was a 5 minute game, and at the point you guys are talking about (over move 40), we really didn't have much time left at all. That's no excuse of course - I still should have seen moves likes 42...Qxg4. 

At that point though (past move 40), I was really just trying to win on time. My opponent had maybe 10 seconds to my minute. What I should have done was take my time (I have 50 seconds more after all), but instead, I moved just as quickly as he was moving and that caused me to make bad moves and allow that two rook checkmate.