I recently acquired some partially-used yarn with no label. There are two different ways to find out what the fiber content of mystery yarn is: the bleach test and the burn test. While the bleach test can only tell you if your yarn is natural or synthetic, I have read that the burn test can actually narrow down the specific type of fiber, based on how the yarn smells and smokes when you burn it. Unfortunately, I don't have a backyard in which to do the burn test, and I don't want my kids to see me making fire inside. They might get ideas...
The bleach test is very simple. If the yarn dissolves in the bleach, then you know it is a natural fiber. If it doesn't dissolve at all, then you know it is a synthetic. If only part of your yarn dissolves, then it is probably some sort of blend.
Get a glass bowl (because you don't want bleach in the pores of your plasticware), and pour an inch or two of bleach into the bowl. Cut short lengths (two inches) of the yarn you want to test. To make sure your results are accurate, also cut a piece of a yarn that you know for sure is wool. If the wool doesn't dissolve, then you will know that something is wrong with your test. Use a tweezers to set the yarn in the bleach. (If you use your fingers, just wash your hands right away.)
I had three mystery yarns to test, plus one yarn that I knew was wool. After 15-20 minutes, the wool began to dissolve. After an hour, there was hardly anything left of the wool. In the pictures below, the wool is the green in the center (dissolved into a few pieces). Since the other three yarns were unharmed, that told me that they are some kind of synthetic.
At some point in the future (when I acquire a backyard), I would like to try the burn test to determine exactly what kind of synthetic they are.