These are the ones I use the most. They consist of an outer waterproof shell and usually have a fleece interior. there is a pocket opening in the front or back to allow you to put in the absorbent inserts. The fleece lining wicks moisture away from babies' skin, leaving it dry to the touch, and you can put in as many or as few inserts as you want making it completely customizable. Common Pocket diapers are Fuzzibunz & BumGenius.
Pro's: The fit is great. They usually come in a one-size system so you can use the same diapers from birth to potty training. They are easy to clean and dry quick. They are waterproof, so you don't have to worry about using a cover.
Con's: You have to pull the inserts out of the diapers, which can be gross for some people. The lining is generally made with fleece which is a synthetic fabric.
AIO cloth diapers are shaped very much like disposable diapers with an hourglass shape and elastic in the legs and waist to keep messes inside. All-in-one cloth diapers consist a waterproof outer, an absorbent soaker, and a one-piece inner. They close with snaps or a hook and loop system similar to Velcro. All-in-one diapers come in different sizes to fit from birth to potty learning. AIOs are one-piece diapers that do not need a cover. This makes them one of the most simple types of cloth diapers. Common Brands are Little Smudgeez, Bumkins and Kushies.
Pro's: They are easy to use. They are one-piece, and great for babysitters or dads.
Con's: They take longer to dry, the fit can be bulky and you need different sizes just like with disposables.
AI2 cloth diapers are similar to AIOs except that the soaker in an AI2 is separate from the diaper body. Usually AI2 soaker is snapped to the diaper body, but it may be laid or sewn in also. AI2s do not need a cover because the diaper body has a waterproof outer layer.
Pro's: They are easy to use, Body and soaker separate, so they dry faster than AIOs, and the soaker can be replaced if needed without buying a whole new diaper.
Con's: fit can be bulky depending on the diaper, and you have to snap in the inserts.
Hybrid diapers are an AI2 system that can use either a disposable insert or a reuseable insert. The insert typically lays or snaps into the waterproof outer shell. They are available in a one-size system and a sized system. Common brands are the Flip, gDiaper, and grovia.
Pro's: They are good for people who want to be greener but don't want to take the full leap into cloth diapers. The disposable inserts are usually flushable and cleanup is easy. They usually fit well and can be used from birth-potty training if you get the one size system.
Con's: The disposable inserts can be just as expensive or more expensive than disposable diapers, so if you are wanting to save money, this is not the way to go. (unless you use the reusable inserts.)
Prefolds with covers
Prefolds are your grandma's diapers. They are a square of cloth that is folded and fastened using pins or a snappi. They are made with cotton, and require a cover.
Pro's: Prefolds are the most economical diapers. They are much cheaper than the other diapers.
Con's: They require a lot of work. you have to fold and pin them which can be difficult on a squirmy baby. They require a cover. you don't really get a great fit, and they can be a bit bulky depending on the size of the baby.
Well, Those are the types of diapers I can think of right now. If you want me to add anything, just let me know! Hope this helps!