In the Dark Night, the Joker, played by Heath Ledger, tells a few different stories about how he got the scars on his face. One story is that his father gave him and his mother similar scars on their faces to make them smile. Another story he tells is that the scars are self-inflicted via a razor blade. He starts to tell Batman a third story of the scars but never finishes.
Even in the comic books, the true story of the scars is never revealed. Some say that the Joker lies on purpose to keep people guessing and maintain his mysterious persona. Others say that even the Joker himself does not know how he got his scars.