What are the types of Black and White film developers?

Forgive me if I don't provide the answer you're looking for; I'm never certain of the intent when the Word "type" is used.

I could answer "powder" and "liquid" or "high contrast" and "normal", or perhaps "types" means manufacturers to you. How's this...

Kodak D-76, Microdol-X and T-Max; suitable for Plus-X Pan or Tri-X Pan

Ilford ID-11, Ilfotec DDX, Microphen, Perceptol; suitable for Ilford Pan-F Plus

Ilford FP4 Plus (ISO 100); HP5 Plus (ISO 400)

Agfa ISO 100 films: Refinal, Rodinal, Rodinal Special

Contrast, as referred to above, is a characteristic of the developer to a degree, and may be modifiable via dilution in some developers, often with a corresponding affect on acutance. These, by the way, could be added to the list of types above as some might consider high acutance or high contrast developers as types.