This is all you really need to know about rice.
Rice (oryza sativa) is a grain, like wheat or corn. There are over 120,000 varieties grown all over the world. But there are two basic types: long-grain, and medium- or short-grain.
The long-grain rice has less surface starch and cooks up light and fluffy, and the grains don't stick together. It's best for pilafs, "Spanish" (Mexican) rice, or soups. Basmati is by far the most comon variety.
The medium- and short-grain rices have more surface starch, and are used in oriental dishes, recipes calling for sauces, and creamy dishes such as Paella and Risotto. The shorter-grain rices are used in sushi and oriental "sticky rice" and include Jasmine and Calrose.
Wild rice is not a rice at all, but a grass from Minnesota. It's expensive and doesn't taste very good, but it's highly nutritious.
Brown rice is white rice which has not been polished, and retains its bran coat and germ. It's more nutritious than polished white rice, but takes a lot longer to cook.
Converted rice is a preprocessed (parboiled) rice. The starch is infused into the kernel, so it cooks fluffier (and to less volume), but not in less time than raw rice. It's also parboiled before milling, so it retains more nutritive value.
Instant rice is precooked, then dried. It cooks slightly faster, but tastes awful. People who hate rice were probably fed this as kids.
If you're not going to steam the rice (as for sticky Japanese sushi rice) you will get improved flavor if you sauté the rice prior to cooking. This also allows the grains to separate and absorb more liquid, useful for most dishes. Rinsing the rice before cooking will make it fluffier and less sticky.
Authentic Paella calls for a white medium-grain rice called Bomba from Calasparra in the Murcia region of Spain, adjacent to Valencia. Since Bomba is so expensive (even for Spaniards), restaurants usually use a more common rice such as Balilla or Sollana or one of the fine rices from the Cullera and Sueca regions of Valencia. Bomba is considered the best paella rice because it absorbs more liquid and becomes plumper in less time than the less expensive rices, probably because it matures at a much slower rate. Some Spaniards refer to their rice not by name, but by origin, such as arroz calasparra or arroz sueca.
Authentic Risotto calls for one of the similar medium-grain rices from Italy such as Arborio, Carnaroli, Maratelli, and Vialone Nano. Good luck finding anything but Arborio in the U.S. If you can't find it, you can substitute any plump short or medium-grain rice, sometimes called pearl rice. The main characteristics of this rice is that the surface starch becomes creamy during cooking.
Less expensive and more common substitutes for superfine rice include Calrose and Baldo. Calrose is a common domestic medium-grain rice that is good for sushi but can also be used for paella or risotto. However, it will take longer to cook and will absorb less liquid (and thus less flavor) than the imported rices, and will have a softer (less "al dente") bite. Baldo is a short-grain rice common in Turkey and Europe which is a better substitute than Calrose for risotto or paella because it is creamier than Calrose and absorbs more liquid.