I'm guessing you're talking about Ilford products, since they have paper developers marketed with all the terms you mention. Donald's covered the cooltone/warmtone difference, but as to Multigrade, don't get hung up on that name in a developer; it's just a marketing name. Any paper developer will work with multigrade papers. FWIW, similar comments apply to some other developer names. For instance, Kodak's T-Max developer works fine with non-T-grain films, and any film developer will work with T-Max films. That said, there are certain combinations of film/paper and developer that work better than others, but you've really got to get into specifics to discuss that. If you're just starting out, I wouldn't worry too much about it, especially for paper developers. Once you get some practice you'll be better able to appreciate the subtle effects of different emulsion/developer combinations.