I need to start out by saying that today's cloth diapers are nothing like your momma's cloth diapers. (Well, actually you can still get cloth diapers that ARE exactly like the ones your momma used....but why would you want to do that?)
Cloth diapers these days run the gamut from super-simple to "I need a degree in rocket science to figure this thing out." Me, I like to stay with the simple stuff.
I'm not going to go into every type of cloth diaper, but I will tell what I've used and what's worked for us.
There are pocket diapers like these.


They come with an insert, and you can buy separate inserts if you like different kinds. (Like these.)
We've got a number of these types of diapers. They have a waterproof outer layer and an inner layer that helps the baby feel dry. You put the inserts into the the diaper and that's what holds all the pee. Once it's stuffed, it goes on just like a disposable diaper. These diapers come in some really cute prints and colors. I really liked them at first, but then came to the realization (after a lot of trial and error trying to figure it out) that Lady is allergic to the fleece liner (the "stay-dry" part). So I stopped using them on her. I've just started trying them out on June and it looks as though she may have the same problem. My kids are known to have extra-sensitive skin, though. I know a lot of babies that don't have an issue at all with these diapers. You can get these in a one-size-fits-all or you can get them in the different sizes.Another kind is the "dad-friendly" cloth diaper. The All-in-One diaper (AIO). This is similar to the pocket diaper except the absorbent layer (the insert) has already been sewn in between the outer and inner layers. This kind is the most disposable-like diaper there is. They can be a bit trickier to get clean since you can't separate the layers. So some people have a problem with them getting stinky. I've never had a problem with ours, but I don't use them that much. Mostly just for church or if I know somebody else will be changing my child. They now make them with a pocket also so that you can add absorbency if needed.
Here's an example of an AIO:

The kind we use the most is just prefolds with a cover. Prefolds are the "workhorses" of cloth diapers. Here is an example of prefolds:
There are a few different ways to use these. You can pin them on like our parents used to do. You can "snappi" them on using a snappi:
Like this:
These need a cover over them to make them waterproof. And since I don't want to bother with pins or snappis, I just fold them in thirds and lay them in the cover and put the cover on like a disposable diaper. SOOO easy!
I do leave the back part of the diaper open (instead of folding it) on June to better catch her breastmilk poops. And I haven't had to deal with a blow-out yet using cloth diapers and I've got a bunch of stained onesies to prove I dealt with them daily when using disposable diapers. My favorite covers are Bummis Super Whisper Wraps. By the way, prefolds and covers are also the least expensive way to go when cloth diapering. :)
I would highly recommend getting a diaper sprayer and hooking it up to your toilet. It's certainly not necessary, but I found it makes my job a LOT easier. You can get one like this, or you can just buy a kitchen faucet sprayer attachment at Home Depot and hook it up to your toilet line. (That's what we did.)
For a diaper pail, I bought a garbage can with a pop-up lid from Target similar to this. And I bought a couple of diaper pail liners like this to rotate. And I bought a couple of wetbags for when we're out and about.
The up-front cost of cloth diapering may seem like a lot, but you'll definitely save money in the long-run.
The reasons I cloth diaper are:
1) My baby's bum is cute now! :)
2) My baby isn't being exposed to yucky chemicals 24/7 anymore. (Did you know that the same chemicals that are in tampons and cause Toxic Shock Syndrome are also in disposable diapers?)
3) I'm not sending hundreds and hundreds of nasty plastic diapers to the dump every year anymore.
4) I'm not spending hundreds of dollars on disposable diapers anymore.
5)It's fun. I know that sounds crazy, but I really do enjoy cloth diapering.
There are some things to know before starting to cloth diaper. You have to change cloth diapers more often than disposable since they're not filled w/chemicals. Once they're wet, they're wet. I don't find it to be that much more frequent, maybe a couple of extra diaper changes a day. No biggie.
You'll obviously be doing a bit more laundry. Cloth diapers add about 3 loads of laundry a week for me. I wash every other day. After the kids go to bed, I throw the bag of dirty diapers in the washer and run it through a rinse cycle. Then I wash on hot and add an extra rinse. Then I pull them out before bed and hang them to dry. You can definitely machine dry them, but I choose not to so that they last even longer. You can't use certain detergents with them (whiteners and brighteners make them less absorbent). And you can't use fabric softener with them. It also makes them lose absorbency. About once every 3-4 months I put a little bit of bleach in when I wash them, just for good measure.
All pictures thanks to bumgenius and www.greenmountaindiapers.com