A psychopathic mastermind who describes himself as an "agent of chaos," The Dark Knight's Joker rises to the top of the criminal underworld by thrusting Gotham into anarchy. Following the events of Batman Begins when Jonathan Crane opened Arkham's doors and released all the criminals, the Joker immediately gets to work with attacking Gotham's crime syndicate. Face sketched into a permanent smile thanks to two large scars on either side of his face, the Joker regularly asks his victims if they want to know where he got his scars, only to change the story each time. In one version, he was attacked by his own father, and in another, he did it to himself after the mob craved up his wife's face.
There could be many reasons for this inconsistency. A glimpse into his ever-changing mind, or perhaps a reference to the character's alternating comic origins. Since his introduction in Batman #1 in 1940, the character has had many origin stories, so many that they were all referenced in Alan Moore's The Killing Joke, implying that the Joker himself can't remember where he came from and that all his backstories are just suggested versions. However, there is a chance that Christopher Nolan had actually given the Joker a backstory hidden in plain sight, as a new fan theory on Reddit claims that the villain may have been telling the truth.
The theory poses the idea that Joker wasn't lying when he told both versions of how he got his scars. He had possibly grown up with an abusive father who had a habit of threatening his mother with knives, explaining Joker's obsession with knives in the film. Eventually, he gets married to a woman that squanders their money away on gambling and other vices. Unable to pay what she owes, Gambol, the underboss in charge of illegal gambling and extortion, carves up her face. This event breaks her and, in turn, sends Joker into madness.
After cutting his own face, he's then sent to Arkham where he spends his days planning revenge against both the mob and Batman, a vigilante who failed to protect the city. The theory goes on to state that, in The Dark Knight, Gambol was Joker's true target. Further evidence cites Joker's early interactions with the mob and how they were all connected to Gambol. Very early in the film, he purposely antagonizes Gambol, leading him to be the first mob boss to die.
However, as nice as this theory sounds, it can easily be nitpicked. The Joker's character has always been a nihilist, he doesn't take things personally and prefers to keep a target on everyone's back. No matter his origins, chaos is always the end result. Never actually telling anyone how he got the scars maintains a level of control over his victims. He knows everything about them, but they don't know anything about him, not even his name.
As for Gambol, he was just cocky. If he never put the hit out on Joker, it's likely he could have lived to the end. Joker antagonized him during the table meet because he kept throwing insults, calling him a "freak" and demanding that Joker give them back their money. Psychotic or not, no one wants to be called a freak. The most likely scenario is that the Joker was bending the truth to keep his enemies on their toes, something the character has become synonymous with.