I'd like to start by saying what a blessing cloth diapers have been to our family. I remember the excitement of pinning a cloth diaper onto Yehoshua for the first time, when he was a few weeks old... and then later the realization that cloth diapering (or changing diapers, period!) was a never-ending process and everyday part of my life. :)
My cloth diapers have been a blessing, first of all, because they were all gifts. I think 6 different people have helped build up my diaper supply (mostly at the baby shower for Yehoshua) by giving me diapers. I was even given some Gerber vinyl covers, which lasted for a long time with Yehoshua and finally wore out with Eliyahu. I have a variety of different thicknesses and kinds of cloth pre-folds (the most basic, cheapest variety of cloth diaper) and have so many that I can wash a full load of diapers and still have some for the baby to wear!
Cloth diapers have also been a blessing financially. I use cloth diapers for a variety of reasons. The biggest reason, I confess, is financial. Even buying the least expensive disposable diapers, and changing the child infrequently, is still more expensive than cloth diapering. If I can avoid spending money every week for disposable diapers (and then spending money on trash pick-up to get rid of them!) we actually do save a lot. (Yes, even including extra water and electric costs and washing machine wear.)
I like the feeling of soft clean cotton against my baby's skin, and being able to change my baby whenever he wets without worrying about how many diapers we're going through. The clean diapers smell so fresh and nice, unlike disposables (which in my opinion stink unless they have perfumes added, which is probably bad for the baby!). I don't mind doing the extra laundry; it's a nice feeling to have a basket full of clean dry diapers, waiting to be used again.
And, it's nice to know that I'm not creating a lot of waste. The average child goes through 7,000 diapers before they're potty trained. That's a lot of waste when it's all in disposables in a landfill. I like washing my diapers. The "waste" goes down the drain and the trashcan doesn't stink. (At least, not from diapers. :D) And can I tell you how nice it is to not need to empty the trash every night because it's smelling up the downstairs?! Better yet, I don't even need to take out the trash anymore because it's all either recyclable, compost, or burnable! :)
And, too, the age-old arguments of the convenience of not having to buy big packages of diapers, and never "running out" of diapers (though there have been times where I was so far behind in things that I was running out of clean ones :D).
I think cloth diapers are more work than disposables. So I'm not going to argue that they're easier. In some ways, yes, they are. But not in every way. For us, the benefits and blessings have outweighed the extra work (which isn't maybe as much as you'd imagine).
I enjoyed using cloth diapers with Yehoshua, and planned to use them with Eliyahu as well. With both children, I have used disposables for the first few weeks, particularly at night, while I was adjusting to new schedules and recovering from the birth. Newborns are so unpredictable (at least, mine were) but after a few weeks, we were settled into somewhat of a schedule.
What I didn't know with Eliyahu (at first, anyway), was that he was sensitive to something (still don't know what, exactly) in disposable diapers. By his third day of life, he was already starting to get diaper rash. It continued despite all my efforts to get rid of it. Someone had given us a package of Huggies diapers, and I somehow realised that when we used that brand, his rash got better. We used Huggies and the rash did start clearing up noticeably within about 12 hours. I tried a number of different brands of disposable diapers, and all caused an awful rash. Huggies diapers, even with coupons, were outrageously expensive, so to cloth we went.
Eliyahu hasn't had a single rash since we started using cloth diapers. Yehoshua only got diaper rash once when he was a baby, and I discovered it was due to some detergent residue in the diapers. I have had such good success with cloth diapering. My house doesn't stink, and I know that my children are in soft, dry, nice-smelling diapers. (I never understood why people used the term "paper diapers" for disposable diapers, until I used cloth for a while and then felt a disposable!)
So there's some back ground for you; just my personal thoughts and why we personally like cloth. Now, for the how-to!
I have a kind of diaper called a "pre-fold [1]". There are a lot of "fancy" (and expensive!) very nice cloth diapers out there. I haven't had experience with them, so I won't be talking about them in this article. (As a side note, I have used flat-fold diapers, and don't mind them, but pre-folds do save me time, so I sewed all of my flat-fold diapers into pre-folds.) Pre-folds are the cheapest, most-durable, and easiest-to-wash diapers. They require some sort of waterproof cover, like the Gerber vinyl covers sold at Wal-mart (which I don't recommend--keep reading).

Here is a picture of some of my pre-folded diapers. At the top left, you see a very thin diaper with a washcloth folded in thirds and laid in the center, to increase the absorbency. At the top right, there is one of Eliyahu's nighttime diapers, which is really just two thinner diapers with a washcloth folded inside. The bottom right diaper is a fairly thin daytime diaper for Eliyahu, and the bottom left is Yehoshua's daytime diaper, a medium-thick one.
You can probably see that there isn't really one right way to "do" cloth diapers, at least with pre-folds. I take whatever I have and make it into the absorbency I need. Washcloths are great for increasing the absorbency without increasing the bulk too much.
Here is how I fold my "pre-folds":



This much folding would be good for a very small baby. A larger child would need very little of the front folded under (step 3). The diapers can be folded to fit whatever size baby you have, which is one of the reasons that pre-folds are more economical than fitted diapers, where a variety of sizes are needed.

Here are some diaper covers. These are the four different kinds I have. There are other, "fancier" covers with which I haven't had experience, so do your research and check your budget and make choices based on what's best for you. I'm just writing about what has worked for me. :)
At the top left, you see a diaper cover that opens at the sides and has snaps. The snaps are very nice, and adjustable (so it fits while the baby grows!), and there's some airflow at the sides because of the snaps. These are all very good features. Side opening = no poop on baby's legs. Adjustable snaps = fits baby longer. Airflow at sides = less chance of rashes. The only slight issue with this cover is that sometimes during a long nap it will leak, especially if the diaper inside isn't as absorbent as it should have been.
At the top middle, you see a Gerber vinyl diaper cover. These can be found at Wal-mart, and they are cheap for a reason. They do work well, but they don't last long at all. If you buy these, you can almost plan on replacing them in a couple months' time. I have found that when these covers crack, plastic tape can be placed over the cracks and will go through the washer and last for a while longer. I don't throw mine away until they are beyond tape-repair. :)
At the bottom of the photo, there is a velcro cover. This cover is vinyl, with cotton cloth on the outside. It has velcro, and elastic around the waist, so it's adjustable and comfortable. There are a lot of great things about this cover (like the first cover mentioned) but a few drawbacks. The cotton on the outside wicks moisture from the edges of the diaper. Velcro wears out a lot faster than snaps, even when it's washed correctly. And this cover tends to leak during naps or nighttime use.
At the very right is my favorite diaper cover. I purchased these [2] nylon diaper covers when my vinyl ones all wore out in the smaller sizes. I can't say enough good things about this cover. It doesn't leak. It's still fairly soft. It's machine-washable. It's affordable. It doesn't wear out. If I have to buy more covers, I plan to get this kind. At $2 each, they're much cheaper than the very nice ones.

And the last of my supplies: cloth wipes. If you're washing diapers, you may as well wash wipes, too. And it's so much cheaper and nicer than buying the commercially made ones laden with ingredients I can't pronounce. Top left is a plain white washcloth, top right is some inexpensive baby washcloths. Bottom right is some homemade washcloths which were a gift to me, and bottom left is unhemmed wipes cut from old baby bath towels and socks.
You can also make homemade baby wipes using Bounty paper towels, boiling water, a little baby bath soap, and olive oil. I have made those in the past, but I find it simpler to just use washcloths with plain water.

Here is what the diaper looks like when it's pinned on. I prefer regular pins because they hold well. (I've never poked the baby, though I have poked my own finger a couple times!) I do have a snappi [3] but it won't snag most of my diapers and I'm so clumsy at using it anyway. I know others who love snappis, so I think it's a matter of preference. :)

Here is the diaper, covered with a Dappi (brand) nylon cover. And, the happy baby, all snapped up :).
So, my basic supplies are: cotton pre-folded diapers, nylon diaper covers, and a pair of diaper pins.
Optional supplies for me include rectangular fleece liners (cut from an old blanket) which can be used to line the diaper so the baby feels dry; I don't mind using them but found that they were more bother to hang on the clothesline than they were really worth.
Now for the exciting part: washing and drying cloth diapers.
I want to be sure to mention that cloth diapers is just another part of parenting, and your method will probably depend more on your circumstances, such as your house or lifestyle, rather than a certain "right" or best way. I have changed how I do things several times. Here is a basic overview:
With a baby who is exclusively breastfed, diapers can just be thrown into the washer. Do a pre-soak/rinse, and then your normal wash cycle with an extra rinse. I did this with Yehoshua, and it saved me a lot of time (rather than washing out all the poopy ones). With Eliyahu, he likes to fill his diaper only once every day or two, so I wash his out in the toilet since they're so full and far between. :) With a child who is eating solid foods, diapers need to be rinsed/scrubbed out in the toilet.
I have always done a "dry pail" which means I don't keep any water in my diaper pail. I do sometimes toss some baking soda in the pail with the diapers, but not always.
The night before I plan to wash the diapers, I put them in the washer and start it (cold water). I leave the lid up so it washed and then just soaks. The next morning, I continue the wash, adding a small amount of detergent and some baking soda. Then I do a second wash/rinse (warm/cold) with plain water (no more soap). This gets the diapers nice and clean. If I remember, I add a couple tablespoons of white vinegar to the final rinse water, as a natural fabric softener and to combat hard-water smell.
And lastly, if at all possible, I hang the diapers (and covers!) out in the sun. Any stains on the diapers magically disappear after a few hours in the sun, leaving bright white diapers and somehow also helping combat the likelihood of diaper rash. Sun-drying diapers is the way to go if at all possible.

This is the usual state of my diaper supply: a laundry basket full of unfolded clean diapers, and a pail of dirty ones. I love to have my diapers all folded and stacked neatly. With both children still in diapers, I use about 25 diapers + 10 wipes/washcloths/liners each day. I have about a 3-day supply of diapers. All my efforts at folding and neatly stacking are used up within 48 hours. So for now, I just store them in the basket and grab what I need. The exception is that I do fold nighttime diapers, so I can change Eliyahu during the night with minimal effort. (I change him on the bed in the dark!)
Combatting Diaper Rash
Here are some things I have found useful for keeping diaper rash away:
1. Change baby frequently. Check for wet diapers all the time, and change as often as needed!
2. When changing a diaper, gently pat (not rub) the diaper area with a soft dry washcloth, removing any excess moisture. Allow baby's bottom to completely air-dry between diaper changes; don't just stick a new diaper on right away. Give baby a toy to occupy him during this length of time :D
3. Switch covers. Diaper covers that were used for wet diapers don't need to be washed after every use. However, they should be air-dried between uses. I usually have 2-3 covers "in rotation" so I always have a dry one to put on the baby.
4. Make sure you use very little detergent when you wash your diapers. Very little. Open your washer and look to see if there are soap suds in your final rinse. You might be surprised at how far a little detergent goes.
5. Don't use perfumed detergent. Don't use fabric softener (ever!!).
6. Give your diapers as much sunlight as possible.
7. Use only water when you clean baby's bottom. Be sure to air-dry after you've washed baby's bottom AND when changing wet diapers (which don't require wipes). Allow baby's bottom as much exposure to the fresh air as is possible. My diaper changes usually take 5-15 minutes.
8. If the rash is really bad, lengthen the amount of fresh air it receives. Consider using a little rash ointment for your baby's comfort. (Try not to get the ointment on the cloth diaper, though, as it's nearly impossible to wash out!) Switch to disposables if needed, so you can use a more liberal amount of ointment without ruining your cloth diapers.
This has gotten long, and I'm sure I haven't covered everything! If you have questions, please leave comments and I will either reply to your comment or edit my post. There are also some great websites about cloth diapering, such as diaper safari [4], the diaper hyena [5], and diaper pin [6].
Links:
[1] http://www.greenmountaindiapers.com/diapers.htm
[2] http://www.babybestbuy.com/diapering-nylon-pants-2.asp
[3] http://www.babycottonbottoms.com/fasteners.htm
[4] http://www.diapersafari.com/
[5] http://www.thediaperhyena.com/
[6] http://www.diaperpin.com/home.asp