It depends on the plant. Most plants, especially succulents, will work with cuttings (clonings, if you prefer that term). Roses and other more "tree-like" plants would be propogated by grafting them onto another, existing, plant. Same with fruit trees.
Porthos ivy, African violets, and "Chicks and Hens" are some of the easiest. Heck, my mom's Christmas cactus always manages to invade any pot that one of its leaves wanders into. Small plants tend to have little off-shoots that are easier to harvest than just a leaf, but almost every common plant I know of can be reproduced using a cutting.