aka, Is This a Diaper or a Cover?
The first step in getting a handle on the world of cloth diapers is working out what is what. But terminology by itself only gets you so far, so I'm including some info on the differences and when you might choose each of these options. As usual, this is all my opinion, and what works for me might not work for you!
Diapers that need covers
Prefolds - These are the classic that most people picture when you say "cloth diaper." Several layers of cotton folded on top of each other. I will be honest, I know there is Indian and Chinese cotton, but I've never gotten the differences straight and I have no clue which I own. Not convinced it matters. These require a cover, as the pee will soak right through them. You can lay them in the cover then put it on the baby, or fold the diaper and use pins or a snappi to put it on the baby and then put the cover over it. I find that the latter gives me less leaks, but others swear by the former. Prefolds are cheap - as little as a dollar each, though commonly more like $2.
Fitteds - A fitted diaper is like a prefold, but is precut to fit the baby with no folding, has elastic in the legs and waist, and closes with snaps or aplix (a version of velcro common on cloth diapers). They are also often made with a cool fabric on the outside layer, which leads to some brands (ie, Goodmamas) being highly sought-after as collectible. Like prefolds, these require covers; there is no waterproof layer in a fitted. Personally, I don't really get fitteds. They are far more expensive than prefolds - you might get a few for $5 each used, but new they're anywhere from $10-$40 each. And then you have to put a cover over the cool pattern anyhow! But again, some people swear by them. They cite the convenience and the elastic holding more poop in to keep covers clean longer, and many people enjoy the patterns even though they get covered up.
Contours - Like prefolds, but shaped to fit the baby without extra folding. Both conceptually and price-wise in between prefolds and fitteds. These do require a cover. I really don't see these mentioned much on the cloth diapering boards I read, so I get the feeling most people go fully to either the prefold end or the fitted end rather than taking this middle ground.
Covers - You'll need to put one of these over any of the above types of diapers. A cover is not a diaper itself, and generally has no absorbent material, only a layer of waterproof material to go over the absorber of your choice. Many commercial covers use PUL as the waterproof material, but there are also covers that use fleece or wool. You can usually use one cover repeatedly until it gets poop (or a lot of pee) on it. They range in price from about $8-20 each new, and for a newborn 5-8 of them will get you through laundry every other day. They come in every color and pattern possible, especially if you start looking at handmade ones on Etsy or Hyena Cart, and close with either snaps or aplix.
Diapers that don't need covers
Pockets - A pocket is a diaper that does not require a cover. It has the same kind of waterproof material (generally PUL) on the outside as a cover, a minimal layer of absorbent material on the inside (sometimes a material that wicks moisture away from the baby's skin), and a pocket between them to put more absorbent material into in the form of inserts, doublers, or even a trifolded prefold. Personally, I find stuffing pockets - then having to pull out the wet inserts - to be annoying enough that I've sold the few that I owned, but plenty of cloth diaperers use them exclusively. These are generally more expensive than prefolds + covers, though there are cheap brands sold mostly on eBay.
All-in-Ones - As the name implies, this is the whole shebang. A waterproof outer layer with all the absorbent materials you need attached. AIOs are the most expensive option, but many people keep a few around for use by babysitters or family who don't want to mess with snappis or pocket-stuffing; using an AIO is just as easy as a disposable. I like them better than pockets, but only have a few for overnights (with hemp doublers) because of the cost. Because of all the fabric, they often dry more slowly than other diapers.
All-in-Twos - What? Wouldn't that be a pocket? That was my initial reaction to the existence of AI2s, but the difference seems to be that the material snaps into place rather than going into a pocket. So I guess you can pull it off for faster drying, while not having to fish anything out of a soggy pocket.
Other bits and pieces
Inserts - These are what go into pockets. They are long, thin rectangles of absorbent material; microfiber, cotton, hemp, and bamboo are most common. Generally one will get you normal absorbency and two will do overnight duty. You can also lay them into a cover instead of a prefold - but remember that microfiber shouldn't touch a baby's skin directly and can cause rashes if it does.
Doublers - Are basically the same as inserts, though they may be thinner. The point of a doubler is to lay over another diaper to increase absorbency for overnight or heavy wetters.
Liners - These are there to act as a barrier between the diaper and something. You might use liners if you need to use a non-cloth-friendly diaper cream, or to make it easier to shake poop off into the toilet. There are both flushable paper and fabric liners; if you are washing liners to reuse that have diaper cream on them, don't put them in with your other diaper laundry.
Soakers/Longies/Woolies/Skirties - Various names for wool covers. Longies are pants-length, and skirties (duh) include a skirt. I honestly don't know much about how wool works; it's not actually waterproof, instead it somehow lets the moisture evaporate out slowly instead of leaking out. (I think.) Plenty of people swear by it even though a single soaker can cost $20-50.