Types of Cloth Diapers

Flats: Literally a piece of fabric that you fold around the baby origami style. The only ones I have are Gerber Birdseye that I use to stuff pockets but I wouldn't recommend these for actual diapers. Walmart sells flour sack towels that apparently are great for this purpose.
PRO/CON: There's a billion different folds so you can customize these for whatever you need however, you have to learn the folds and then teach them to your husband or babysitters
PRO: they can be used throughout diapering because they are generally one size and you just modify the fold as they grow.

PRO: you can stuff your pocket diapers with these
PRO: they dry super quick
PRO: Cheapest way to go
CON: need a separate cover
CON: not good for older babies(really over 3 mo) or squirmy babies
CON: not the best for heavy wetters

 

Green Mountain Diapers come in lots of sizes!

Prefold: A flat with an absorbent material sewn in "the wet zone". We have a ton of these and my husband really liked these. They come in sized or one size/toddler. Green Mountain Diaper prefolds are really great.
PRO: less folding than a flat but still customizable
PRO: nearly as cheap as flats
PRO: can be used to stuff pockets later
PRO: better absorbancy
CON: still folding involved so squirmy babies are difficult to pin down
CON: Need a separate cover

For baby #2!

Fitteds: A prefold with elastic sewn into the legs and back to make it conform to the baby. some have velcro or snaps, others don't. I have a bumgenius and several Thirsties Fab Fitteds that I like. I really want to try kissaluvs for the next baby but I have not bought any yet.
PRO: No folding but similar absorbancy to a prefold
PRO: Great resale value
CON: these tend to be stupidly expensive, I don't understand it
CON: can't be used for another purpose once they outgrow them
CON: Need a separate cover

Thirsties cover! I want this print!

Thirsties cover! I want this print!

Covers: These are used over a flat, prefold or fitted to provide a waterproof barrier made of a fabric called PUL. I was given dappi covers that are ok. I like them over fitteds but don't often use them. Also, I have Thirsties Duo Wrap and Bummis Super Whisper Wrap. They are both SUPER popular for good reason. Get both, you'll be happy you did.
PRO: they come in CUTE prints
PRO: If it's just pee, just change the prefold inside and not the cover. Saves on laundry
CON: You need absorbancy inside them as they are no good on their own
CON: for little babies, poop almost always gets on the cover which means:
CON: you'll need several for a newborn, less for an older child
**I should note that some people use what is called a wool or fleece "soaker" in place of a cover. They are both somewhat waterproof but also come with their own issues like compression leaks. I don't recommend these but you might try them to see if you like it. Some people swear by them.**

This is a Sunbaby

Pocket: These have the waterproof barrier with a stay dry material against the baby's skin. They tend to be the most popular method of cloth diapering. There's a huge amount of brands on these but I'll talk about the ones I liked and the ones I didn't after I tell you the general pros and cons of pockets.
PRO: relatively cheap ($5-$15 each) but this also means that resale on these isn't great.
PRO: easy on/off no folding around the baby
PRO: they dry fast
CON: they need to be stuffed. Some people HATE stuffing diapers. It doesn't bother me at all
CON: you have to pull out the inserts to wash
CON: when you wash, you have to touch pee sometimes. The insert is really east to get out and you only have to touch the edge but sometimes it's soaked.
BRANDS:
 

BAD: Happy Heiny: I just really wasn't impressed with these. They're pretty cheap and act like a cheap diaper. They leaked pee no matter how much absorbancy I put in.

I have three of this print!

GOOD: Rocky mountain diapers: These are kind of an obscure brand but I love them. They fit the my daughter's big butt well and they are the only onesize that isn't bulky because they have an internal adjustment system similar to the waistbands of kids pants. The snap on the inside spreads the extra fabric out so it's not all in the front.
 

Sunbaby/Kawaii/Alva: three very inexpensive pockets that you can get off of ebay or a swap. Worth it. These are one size.

Fuzzibuns: This is all my aunt and uncle use. I'm not the biggest fan but they're ok. They never leaked and tended to be really trim but I like one size diapers for the most part and although they do make these, I only had size small. They tended to be a little expensive for what I thought you got. (This is only my opinion. They tend to be popular)

 

 

All in ones (AIO): These are the most like disposables that you can get. They have a waterproof outside and an absorbent material sewn inside. The better ones will have a pocket so you can add extra absorbancy but you don't have to stuff them. I have Thirsties Duo Diapers and have tried Bumgenius
PRO: No fuss, touching inserts or stuffing pockets
PRO: good resale value(also more expensive)
CON: They take FOREVER to dry. Most of what I have is pockets and I can't stand how long it takes my AIO to dry!
Bumgenius: the best selling diaper on the market. I don't understand it but people love these things. They did dry faster than my Thirsties but the velcro on the older ones is crap. Apparently they have fixed that. Still not the biggest fan. They also make a pocket that is pretty popular (I haven't tried it. They're a little expensive)

 

Grovia

All in Two (AI2): Similar to an AIO but the absorbent material is only sewn(or snapped into) into one end so when it dries it is much faster. These are generally going to be made by a person but some brands do carry them. Grovia and Flips are two types of these that I have not tried. I have a WAHM(work at home mom) dream eze that is VERY old and falling apart. 

This has similar pros and cons to an AIO except drying faster and in some, the liner can snap out and be changed similar to using a prefold and cover but with no folding.

G diapers look o-so-cute but I'm not sold...

Hybrid systems: They are a waterproof cover and a biodegradable liner. I have a friend who swears by them but the just seemed crappy to me. Brands are Gdiapers and Flips.
PROS: less laundry
PRO: you can use these with a flat or prefold instead of the liner (some have liners you can buy)
CON: generally more expensive and resale isn't great
CON: buying liners can get expensive not sure of savings here...

When I was starting out, I had a friend lend me her diapers to see what I liked. I'm really glad she did because I was able to try a few different kinds and see what I liked. Buy used if you're not sure what you'll like. Babycenter has a great swap site that I frequent but I know others exist.