1. Chrome Plated Steel
The first set of sockets most people get is a 40 piece hex socket set. These are also known as the 6 point socket. There are 12 point socket also but they are still used on the six sided nut and bolt. These kits come with a ratchet wrench and 20 SAE (inches) sizes, and MM (millimeter) metric sizes. Both types fit over six sided nuts and bolts, the common ones found around the house. They usually are made of steel plated with chrome to keep them from rusting. They are very serviceable for things in a home, and may be all you may need if you're not going to branch out to other types of work.
2. Higher Grade Chrome Vanadium Steel Alloy
One step up from this set (#1) would be the hex socket set described above, but is made from a hard chrome vanadium steel alloy. This is much tougher metal and is more resistant to rusting and is stronger. They are usually polished to a mirror finish. This type of kit will last you a lifetime.
3. Non- Hex sockets
If you get into things outside of your home you will find you run into nuts that are not six sided. Automobiles for example usually are five sided nuts and bolts. You need a different series of socket for this type of work. Most non-hex sockets are used industrially.
4. Impact Socket Sets
These kits are made to be used on an impact power driver. They are usually made from chrome molybdenum steel, but you also find them in chrome vanadium steel. Whichever alloy they are made from the metal is softer and can mold to the sides of the nut or bolt. These sockets are made for heavy duty jobs and are more brittle harder than the previously discussed sets, which can be dangerous in power driven situations. The harder ones can shatter if the nut you're trying to get off is stronger than you are. The impact sockets will simple break into two pieces and are safer in these situations. The impact sockets look different and are not plated in chrome or polished to a mirror finish, but are usually black or a metallic gray.