Joker was the Red Hood.
The Red Hood is Jason Todd.
Therefore, the Joker's real name is Jason Todd, QED.

There is no problem that cannot be solved by bad logic.

Even the joker doesn't know. Even in the Countdown series (where they give out the origins of DC characters at the end of each issue), joker isn't sure which among his "origins" is true, if any at all. even his origin is as crazy as he is.

The Jack Napier alias, held over from the Burton movie, also appears in the animated Batman: Mask of the Phantasm. (It's used in a sequence either involving the Batcomputer or Gordon's police files.)

But it's obviously another of Joker's fake names, as it's just a pun permutation of the word 'jackanapes.'