What are these things?
Go ask Webster, for he knows all1.
What's in them?
Depilatories contain a basic substance such as calcium thioglycolate, sodium thioglycolate, or barium sulfide. These chemicals are foul-smelling and corrosive alkalis, though not as strong as, say, lye. They work by dissolving the disulfide bonds in the keratin that makes up hair.
Isn't it dangerous to apply strong alkalis to unprotected skin?
Yes. Depilatories can cause second- or third-degree chemical burns if left on for too long, so be careful. Barium sulfide will burn human skin after 3-5 minutes, while one can apply the thioglycolates for about 15 minutes before the pain starts. Don't use these on thin or sensitive skin, especially that of the face.
Hang on...Aren't fingernails made out of keratin?
Yes indeed, and bravo for catching that. Be sure to wear rubber gloves or wash your hands very thoroughly when using depilatories.
These seem dangerous. Why would anyone use such a thing?
A. Some people will inject themselves with lethal poisons for vanity's sake, and B. These dissolve hair down to the follicles, so the effects last a bit longer than shaving. They're also less painful than waxes and less permanent than electrolysis. When used properly, there's little to fear from depilatories.
So how do I use them properly?
RTFM; it should be on the package somewhere. Test them on a small patch of skin before rubbing them all over your body. Take a hot shower or bath and scrub vigorously after use to prevent ingrowth of the removed hairs. Don't use them on inflamed or broken skin and consult your doctor if anything goes awry.
1. Webster's etymology is off. "Depilatory" comes from the Latin De, meaning "off" or "out of", and pilus, meaning hair.
Sources:
http://www.skinbiology.com/hairremovalmethods.html
http://www.consumerbeware.com/depilatory.htm
http://emedicine.com/derm/topic354.htm
http://www.fda.gov/fdac/features/796_hair.html