After a blunder, is it better to resign or keep playing? I often wonder if it is better to resign or keep playing at a disadvantage after making an obvious blunder (for instance, leaving a piece hanging, only to realize too late that you made the mistake).

Let me be clear: I am referring only to casual games (not tournament), and my main goals are 1) to have fun and 2) to improve. 

It can be frustrating to keep playing at a disadvantage, but does that additional challenge help make you stronger, or is it better to just resign and start a new game?

An example would be the game below, where I blundered (black) on move 13 and left my bishop hanging... should I keep playing or resign?

 [FEN ""]

1.e4 e5 2.Nf3 Nc6 3.Bc4 Nf6 4.Ng5 d5 5.exd5 Na5 6.Bb5+ Bd7 7.Bxd7+ Qxd7 8.Nc3 Nxd5 9.d3 Nc6 10.O-O Nxc3 11.bxc3 O-O-O 12.f4 f6 13.Ne4 Bc5+
First, I would try and keep a very straight face, and maybe your opponent will "trust" you, and overlook the blunder. It has happened before even with very strong players.

That said, if the opponent takes it, you should probably decide based on your level of skill and your perceived level of skill of the opponent, and if the position is messy.

For example, if you are playing a master, and the position is just a clear piece down, you should probably resign. If you are playing an 800-player, then it is very possible that you can play on despite your error.

What I mean by "messy" is if the position is a tactical mess even without the piece, it is more likely that your opponent can still make an error, but if it is just a clear piece down with no tricks, as in the position you gave, then you may as well resign if your opponent is a reasonably decent player.

Lastly, is it a blitz game, or an over-the-board tournament game? (I know in this case, you said it was casual.) If it is blitz, it really does not matter as much either way since it will be over soon; but if it is OTB, it matters more, but your opponent also has that much more time to figure things out.

In the end, there is no right or wrong answer, and you have to evaluate the circumstances, and do what you think is best at that moment.

Do keep in mind that if it were a tournament game, that playing it out may also tire you for the next game.