(Inspired by my seeing the new Batman: Mystery of the Batwoman DVD last night.)
This is possibly the greatest action cartoon of all time, in most of its incarnations. (Teen Titans sucks pretty hard, but everything else rocks!)
First, on the heels of the films, there was Batman: The Animated Series. This defined a new, abstract design for cartoons that continues today in series like JLA and Masters of the Universe (despite the overall increasing trend toward anime). Taking visuals from the first two movies and Frank Miller's work, the series created a totally new continuity that, in many ways, was better than the comics themselves. With top-notch voice talent (Mark Hamill, Richard Moll, Kevin Conroy) and stories that were far grittier than any other mainstream cartoon to date, adults liked the show as much as (or even more than) their kids. The show even aired in primetime for a short while.
My personal favorite episodes are "Almost Got Him" (in which various villains describe almost killing Batman, and Batman disguises himself as an appropriately dense Killer Croc) and "The Laughing Fish" (mainly for the sidesplittingly funny television commercial produced by the Joker).
Animated Batman made his big screen debut in Batman: Mask of the Phantasm. Despite doing poorly in the theater (due to its unexpectedly adult content) and dooming all future films to video release, this remains the best Batman film ever. The Joker gets his butt whomped on (with blood and teeth flying everywere), and Batman gets his groove on. All set to a great score by Shirley Jackson that's right at home with Elfman's Batman scores.
I believe next was the switchover from Batman: The Animated Series to The Adventures of Batman and Robin. This was set a few years in the future, with Dick Grayson now Nightwing and a young Tim Drake as the second Robin. (Jason Todd never existed in this continuity, although this Tim's origin is basically Jason's comic book origin.) Batgirl began appearing more regularly , and the animation style became much more cartoony. Most characters were redesigned, all for the worse except for the Joker and Mr Freeze. Sadly, by this point, my schedule prevented me from seeing most of these episodes.
Next came Batman: Sub-Zero. Featuring a tale of Mr Freeze vs Batgirl, this was obviously created to cash in on the "craze" of the craptacular Batman & Robin movie. Despite some rather clunky CGI effects, this one was pretty good, but no Mask of the Phantasm.
Superman wasn't exactly a Batman show, but employed many of the same creators. Batman met Superman in the creatively titled The Batman/Superman Movie, in which Joker and Luthor team up. Another solid film. (And from the little of the Superman series I saw, another solid series. Especially since the Main Man himself, Lobo, showed up! )
Batman and Superman team up on a more regular basis in JLA, also starring Flash (Barry Allen with Wally West's personality), Wonder Woman, Martian Manhunter, and Green Lantern (John Stewart). Still airing on Cartoon Network, but I think maybe a total of six episodes have been produced. Very good show, but I think every time I turn it on, I see the Gorilla Grodd or Aquaman episodes.
Teen Titans features Robin, but that's where it ends. Let's just forget I mentioned it.
Mystery of the Batwoman, which came out last week, is yet another great movie. Someone is wearing a suit modeled on Batman's to cause some pain to the Penguin and Rupert Thorne, but who is she? (Hardcore Bat-afficianados will catch a clever homage to the secret identity of the Golden Age Batwoman, which I won't mention here so as not to spoil.)
This movie ends with the mystery unravelled (duh), and the supposed death of Bane, but we know that can't be the case. A few years earlier, Batman Beyond, set years in the future with an elderly Bruce Wayne acting as mentor to the new Batman, showed Bane's true final fate. A different feel in this series, which isn't lousy like most other "take a familiar character and put him in the future" shows.
And the final step (chronologically) is the excellent Batman Beyond: Return of the Joker movie. Not the best Batman animated movie, but definitely the darkest. In fact, Warner initially censored it for release, then released the uncut version when fans complained. The fate of Tim Drake is enough to make you wish he'd gone out the same way as comic-book Jason Todd (minus the Lazarus Pit, turning to evil, and all that other stuff from the Hush storyline) just to make it easier on him.