I have Ilford HP5 Plus 400, I believe this is one of the standard B&W films. I've developed this myself, but I'm not sure what to look for or ask a lab tech to make sure that they develop this properly. C-41 is the standard color equivalent to what B&W process?
Most labs I've seen that do black and white processing call it generally D-76 processing, but they could use any number of other chemicals. HP5 is about a standard a b&w film as you'll find, so if the lab claims they do black and white processing, they'll be able to do HP5.
Ilford's site has a PDF for all the development solutions and times that they recommend if you need something specific to ask about.
If you want to shoot black and white film but can only find places that do color processing (for example, most drug store labs only do color film now), you might want to look into chromogenic films like Ilford XP2 or Kodak T400CN. Those are b&w films but they're designed to be developed with the chemicals for color film.
If you are using Ilford film, like HP5, it's better use Ilford developer like Ilfosol or Id11. I recommend ID11: it results in fine grain and better contrast, and if you are doing color, use Ilford XP2