I can answer almost any question you have, as well, so feel free to ask. The basic rundown is this:
Most inexpensive: prefolds and covers--you can either fold the prefold in a rectangle and lay it in or you can wrap the prefold around baby and secure with pins or a snappi. When you change, you just change the prefold and reuse the cover unless it has poop on a fabric part that isn't wipeable. These are great for the newborn period because they outgrow them so quickly, but we also like them for older stages, too. These still make up about half of my "stash".
Hybrid diapers: These are the same concept as a folded lay-in prefold and cover, but most companies have their own inserts that may or may not be stay-dry against baby's skin. Some companies also offer disposable inserts. gDiapers (my least favorite), flip, grovia, and best bottoms are all hybrids.
All in Twos (AI2s): These have a waterproof outer (pul) and a soft inner with an insert that snaps in. When you change an AI2, you change the whole thing each time. Because the insert snaps in, it flaps loose from the shell in the wash, making it an easy wash and dry. I only have one of these because I don't like how the inserts shift on Nathan, but some people really love them.
All in Twos (AI2s): These have a waterproof outer (pul) and a soft inner with an insert that snaps in. When you change an AI2, you change the whole thing each time. Because the insert snaps in, it flaps loose from the shell in the wash, making it an easy wash and dry. I only have one of these because I don't like how the inserts shift on Nathan, but some people really love them.
Pocket diapers: These have a waterproof outer (pul) and a stay dry inner material (suedecloth, microfleece, microchamois, micropique, or minky). You stuff an insert inside the pocket. When you change a pocket, you change the whole thing. I like to use these when we're out and about because it's a super quick change. My pockets all unstuff themselves in the washer, so I don't bother with removing inserts when I put them in the pail. Pockets are nice because the insert washes and dries separately from the shell so they're easy to clean; I use trifold inserts so they open even further in the wash for a faster dry.
All in Ones (AIOs): These have a waterproof outer (pul), a soft/stay dry inner, and the absorbent part of the diaper sewn in. Changing them is pretty much like changing a sposie--put it on, take it off, drop it in the pail. Because the absorbent part is sewn in, they are harder to get clean (water doesn't swish the whole way through because of the waterproof outer) and they take foreeevvvveeeerrrrrr to dry. I have a couple of AIO trainers and I don't like them, so they're at the very bottom of the pile.
Fitted diapers: These are NOT waterproof, but they are very absorbent. They typically have a stay dry inner and can be made in the style of an aio, ai2, or pocket. When you change a fitted, you change the whole thing. I like to use these during the day without a cover when we are hanging out around the house, changing them when the outside is damp. I love to use them for nights with wool longies or shorties over them as a cover--that combination is bulletproof for even my super-soakers.
Washing: Dry pails are recommended, rather than the wet pails that our parents used. Wet pails are a drowning hazard and a breeding ground for bacteria, plus soaking in water constantly breaks down the elastic and modern diaper fabric fibers more quickly. I use a tall kitchen trashcan with a washable pail liner (I have 2--babies always poop the second you put diapers in the wash!!). Some people use hanging wetbags, and others use 5 gallon buckets that they wash out each laundry day. When I had 2 in diapers, I washed every 36-48 hrs, whenever we got close to running out of diapers. Now with just 1 and a huge stash, I wash every 4-5 days when my pail gets full. I use Tide Original liquid, clean breeze scent. I do a cold rinse, hot wash/cold rinse with tide to line 2 (or to line 3 if the pail was extra full), and then another cold rinse, then put everything in the dryer on low. If you have hard water, I would go with Tide powder because it has water softeners built in.