"My bad" versus "My mistake"

"my bad" (not only informal but also slang) is used to recognize that you're at fault. "I'm sorry" and "my mistake" convey the same.

• my bad a way of admitting a mistake, and apologizing for that mistake, without actually apologizing.

• "You just spilled your beer on my term paper!!" "Er... my bad."

I associate "my bad" with sports. In basketball, for example, someone may attempt to make a quick pass to a teammate, but the ball goes out of bounds. Was it simply an ill-conceived pass, or should the would-be receiver have been paying more attention and made the catch? As both team head to the opposite end of the court, the guilty party may acknowledge to the teammate "My bad."

It's thus an acceptance of responsibility for the misconnection, miscommunication, or whatever— but it's also intrinsically evanescent. If the screw-up were major, "My bad" would be an inadequate and indeed trivializing response. Better form would be to say, "I [messed] up, and that's on me!" or something similarly considered and self-reproachful.

As the poster suspects, the difference between "my bad" and "my mistake" or "my error" is one of register and informality. I've heard people use "my bad" in playground sports in California (where I live) for almost three decades now; in fact, all too often I say it myself (owing to my problematic level of athletic skill). In an in-game milieu, it is not—as the coverage in Urban Dictionary (cited in Centaurus's answer) makes it out to be—a form of "apologizing without actually apologizing." I would say, rather, that in the midst of a sporting event it is an exceedingly brief apology without melodramatic garment rending or waste of breath, but a genuine apology nonetheless.

It may be that in fratboy/bro-speak, "my bad" is used as an utterly insincere acknowledgment of error about which the speaker doesn't really give a rat's ass (so to speak). But in sports it retains its original force as an expression of culpability and regret—and meanwhile the game goes on.

It's used by many as it gets repeated that many times it becomes normal, it's called fosilisation, but in my opinion it's grammatically wrong, that's why it sounds like you're stupid to people that aren't accustomed to it. Bad is an adjective and mistake is a noun, you are supposed to follow a possesive pronoun with an a noun, you wouldn't say... that was my angry, or that was my tired, or that was my courageous, you'd sound like an idiot. You can turn an adjective into a noun for the purpose of abstract, for example the movie... "the good, the bad & the ugly" but making a mistake isn't expressing an abstract quality, you simply screwed up.