Way, way, waayyy too many.
Starting with the easy ones:
- Nolanverse movies, which also includes at least one video game tie-in
- Batman: Gotham Knight, an anime-style animated film set in the Nolanverse, but not canon within it. My best classification would be that it's a parallel universe to the Nolanverse
- Burtonverse movies, otherwise known as everything from Tim Burton's Batman through Batman&Robin (My apologies for reminding everyone that Batman&Robin existed)1
- The DC Animated Universe (DCAU): Batman: TAS, The New Batman Adventures, Batman Beyond, the Justice League shows, Superman, all the spin-off movies, and more.
- The 1960's show and its movie
- All the other Batman shows exist in separate continuities: The Batman, Beware the Batman, Batman: The Brave and the Bold, Gotham, etc.
- The Arkhamverse (Arkham Asylum, Arkham City, etc.), which also include the animated film Assault on Arkham
- Except for tie-in games, the other games all exist in their own continuities
I'm not touching this one; there are about a hundred of them in total. The main ones are the DCnU (The universe of the New52 reboot) and Earth-1 (The old comic continuity). There's a full(?) list here; essentially every time a writer had a cool idea, DC spawned another universe and another continuity. Additionally, every couple of decades DC would take a blowtorch to the tangled mess of continuity snarls and start over; you often hear about the Golden Age and Silver Age comics being separate continuities for this reason.
Such are the oddities when dealing with a 70-year-old franchise.
EDIT: As a final note, it's important to recognize that all of these continuities are interrelated. Obviously non-comic adaptations borrow heavily from the comics, but occasionally the comics borrow from the movies and TV shows as well. There are a few questions on this site about that phenomena (This one about Bane and this one about The Dark Knight spring to mind), and TVTropes has a whole subfolder devoted to Batman on their Canon Immigrant trope page, but some of the more well-known ones include:
- The characters of Harley Quinn and Renee Montoya were both created for Batman: TAS, but they ascended to comics continuity. Montoya has had a few major subplots over the years, including a weird thing with Two-Face, and now she's seems to be turning into a major supporting character in the Gotham TV show. Harley was an immediate fan favourite, was introduced to the comics in the "No-Man's Land" event, and now has her own title comic in the New52 (One of precious few female self-titles, alongside such veterans as Catwoman and Batgirl)
- Mr. Freeze's iconic, tragic backstory was also ascended from TAS. He was originally a pretty generically evil villain (See his portrayal in the 60's Batman show for an example), but the TAS writers turned him into an more complicated and sympathetic character
Similar to the above, many of the voice actors from TAS became so iconic in their roles that they can almost be considered ascended actors. Kevin Conroy, who has practically made a career out of voicing Batman, is the most obvious example, but Mark Hamill (Joker), Arlene Sorkin (Harley), and Tara Strong (Barbara Gordon/Batgirl) have also had long associations with the franchise.
1 As you can see from the discussion in comments, you can make a strong case that Batman and Batman Returns exist in a different universe from Batman Forever and Batman&Robin. The radical shift in tone and imagery (Particularly the style of Gotham: compare the bleak, gothic architecture of Batman to the neon funhouse style of Batman Forever) present a strong case for the two-universe theory, but there are a handful of backreferences to the earlier movies that are hard to reconcile. Your mileage may vary2.
2 An (obviously) unconfirmed fan theory I particularly like (Proposed here) is that they're in the same universe, but Batman Forever and Batman&Robin are in-universe Batman movies. There are admittedly flaws with this theory, but it's overall pretty convincing