The two most common variants I saw at least around Budapest were chicken and veal. I just looked it up on the Hungarian Wikipedia page and they list beef, veal, lamb/mutton, chicken, and fish (usually catfish.) The fish version is obviously cooked for far less amount of time.
The beef version -- I don't know much about that. I'm looking through Hungarian recipe pages and most have sour cream, but some don't. All that "paprikash" means is the adjectival form of "paprika." So "chicken paprikash" means "paprika chicken." According to the Hungarian Wikipedia page, sour cream became part of the recipe in the mid-1800s, under Germanic influence, though there is no cite for that. But I've generally observed the taxonomy to generally mean that a
paprikás is a sour cream-enhanced stew (with the exception of
paprikás krumpli, "paprika potatoes") and
pörkölt being just the generic word for "stew."
The lamb/mutton version is also something I've never seen. It may very well be regional. In Budapest, at least in the late 90s/early 00s, finding lamb/mutton was always a bit of a challenge. I did eventually find one place near my flat that did stock it regularly, but most markets wouldn't have it.
For the fish paprikash, I found a good video recipe
here with English subtitles. It's similar to the regular paprikash recipe (note that he uses the optional tomatoes and also adds caraway to it, which is something I do for goulash soup and beef pörkölt, but don't for paprikash.) The idea here is you make the sauce separately, sear the fish, and add it after the sauce has simmered for about 45 minutes to an hour to finish. Once the fish is cooked through, you remove it, then you add your thickener (sour cream and flour), cook until thickened and the raw flour flavor is out, and are good to go. Serve spooned over the fish.