Thick or thin skin? Strength is one consideration, and so is the sound: thin skins tend not to be as strong and to sound more metallic, thick ones are stronger and sound "drier".
Things to do when taking the old skin off:
photo 
Tip
Choosing a new skin:
Tip Tip Putting on the skin:
photo Gather up the skin hanging down and fold it over the ring into the centre of the drum. Then put the next ring on top, so that the skin is sandwiched between the rings. photo Tip Threading the rope:
; at the top, thread it up through the loop and then back down through the next . When going down at the bottom and up at the top, thread through the loop that already has a strand of rope running through it (not the one that's empty). Tip Then go round the drum again, pulling the rope up and down through the loops, but still not tight, until the bottom ring reaches the drum body. Pulling the skin through the rings:
Work your way round the drum. Tip Tip The first tightening:
Tip How much depends whether you want to have skin folded down over the rings at the end or not. If, like me, you prefer not to, remember that it'll shrink as it dries and, if it's cut too short, may not be held securely between the rings. I've done that a few times, the skin slipped out of the rings when I tried to tighten it to playing pitch, I wasted a goat skin and had to do the whole thing again! Tip
Shaving - the scary bit!
Then start to shave off the hair, working always in the same direction as the hair growth, and towards your body. Tip Tightening the drum:
Tip Tip Reaching and maintaining pitch:
Tip . Bring it back in front of the second one, then thread it behind the first , and pull towards you . The first vertical will slip over the second . Tip
You can use any kind of sharp blade to cut the skin down - knife, razor blade, scissors. I find a good large pair of scissors are easiest to use. While you're doing this, take care not to pull the rings off one side of the drum completely and have to start again! If you've got the bottom ring in place on the bowl of the drum, this isn't likely to happen. If you've not got any spare rope, go back to the start and pull again, a bit harder, taking up all the slack as you go. You should end up with some spare. You can also tie a short piece of rope on to the end, so you can get all the way round. Take it off when you no longer need it. Sometimes the bottom ring ends up skewed. Up-ending the drum on a flat surface and then tightening it that way can help reduce this. But ultimately it's a matter of the fit between ring and shell; how you tighten can only make so much difference. There are other ways to remove the hair than with a blade - you can use depilatory creams, or electric beard-trimmers. I use a razor blade, because that's how I learnt to do it. (Do Brazilian djembistas leave a strip of hair in the centre of their drum, I wonder?)
There are several tools available to help with pulling the rope. I use a gadget called a "clam cleat", from a sailing shop, which grips the rope and allows you to pull very hard, without wearing holes in your hands! It also helps to sit astride the drum. First time round the drum it's obvious which are the next two uncrossed verticals. But, when you're working on a higher row, it's not so obvious. Look at the top ring - it's always the next two verticals there that you have to cross. DON'T leave the skin soaking for too long - it will stink and eventually even be weakened, so much so that it may fall apart when you try to use it! I've found that, if the centre of the skin has about a centimetre play when I lightly tap it, that's OK. If, on the other hand, you want to leave the skin long and folded down over the rings, the way to stop it sticking out from the drum when it dries is to fold it down when it's wet (one or two vertical slits cut in it will help), then hold it in place while it dries with a loop of rope running round the drum. You're unlikely to get your djembe to playing pitch just by hand-tightening the verticals. To do that, you'll need to use the "Mali weave." This is also how you maintain pitch later. What do you do when you've got to the end of one row and want to begin another? You'll find there aren't two adjacent uncrossed verticals. Re-use the second vertical of the last pair as the first of the next. Don't forget - look at the top ring to see which should be the second of the pair. While the drum has no head, you might want to treat the shell, which helps to keep it in good condition. You can use African "shea butter", or linseed oil. Note that if you use oil, it can take a couple of days for it to be fully absorbed.
Reskinning a djembe is a complicated process, especially when you do it for the first time. It's also hard work, but the satisfaction of playing a drum you skinned yourself makes it worth the effort, IMO. These pages give a fairly detailed guide to the process, which people have successfully followed. But, please remember that there are all sorts of ways to achieve essentially the same result: this is what has worked for me. If you need more help, feel free to contact us and ask. Good luck! pb
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