There are many different kinds, but they can be roughly divided into Western and Japanese styles, both of which come in natural and artificial varieties. Stones are much softer than diamonds, and wear away in use. The residue tends to clog the surface and interfere with the cutting action. Japanese stones avoid this by wearing away quickly, and exposing a new surface. They are used with a lot of water to flush away the residue, and require a lot maintenance to keep the surface flat.
Western stones wear more slowly, but still require liquid to flush the residue away.
They also wear away to some extent, and require occasional flattening. The liquid used with Western stones can be either water or oil. Oil is recomended most often, but I prefer water. Oil makes a nasty mess, which gets on your hands, your work, and in your food if the knife is used for food preparation. You can take a stone that has been used with water, dry it out, and switch to oil.