Batwoman season 1, episode 6, "I'll Be Judge, I'll Be Jury," revealed that Jack Napier is the real name of The Joker in the Arrowverse. While this is a reference that is sure to amuse Bat-fans, it is also a marked contrast from how most Batman stories have dealt with the identity of the Clown Prince of Crime.
The precise origins of the Joker and who he was before he became Batman's greatest enemy are usually kept a mystery. While some stories have hinted at his true identity, the vast majority of writers who have handled the Joker in the past feel that half of the character's power comes from his status as an Everyman figure and his belief that anyone could become like him after one bad day. As the Joker himself noted in Batman: The Killing Joke (ironically, the closest thing the Joker has ever had to a definitive origin story in the comics) "If I'm going to have a past, I prefer it to be multiple choice!"
While "I'll Be Judge, I'll Be Jury" does not reveal any details as to who the Joker once was in the reality of the Arrowverse, it did drop his real name. This occurred early on in the episode, when Kate Kane first learned of the murder of Gotham City's Assistant District Attorney Angus Stanton, while watching a news report. The reporter went on to describe Stanton's career in glowing terms. referring to him as a hero who went after "some of the city's most notorious criminals, such as Jack Napier, otherwise known as The Joker."
The name of Jack Napier is a nod to Tim Burton's 1989 Batman movie; one of the rare occasions in any medium where the Joker was given a definitive origin story. The film cast the legendary Jack Nicholson as Jack Napier - the right-hand of mob boss Carl Grissom and, after a botched assassination attempt at Axis Chemicals saw him falling into a vat full of strange liquid, the Joker. The movie also created another link between Joker and Batman, revealing Jack Napier as the petty thug who killed Thomas and Martha Wayne during a routine mugging.
It is unclear how closely the reality of Earth-1 may follow the Tim Burton Batman movies, given that the first film ended with Jack Napier's death and The Joker is apparently still alive on Earth-1 in the Arrowverse. It is equally unclear if audiences can expect to see the Joker show up at some point in the future to take on Batwoman. While it seems unlikely given how historically protective Warner Bros. has been regarding letting the Joker appear without Batman, wild cards tend to show up when they're least expected and the possibility of Supergirl facing Lex Luthor seemed equally unlikely until it happened.