Here is a list of some (and by no means all) of the stringing media now available and what type of beads work best with it.
Silk - A well known classic for bead stringing, silk thread is most often used for pearls. Some beaders also like to use it with stone beads. Personally, I rarely use silk because I find that it frays easily. However, I know a few people who do pearl bead stringing for a little extra cash, and they use silk exclusively for their pearls. Obviously, silk is a higher quality thread than nylon. So, when you're charging people to hand-knot their pearl strands, it only makes sense to use a good quality thread. You can purchase silk on large spools or on cards with attached twisted needles. It also comes in a variety of colors (white, black, gray, pink, etc.) and sizes (#1-#8).
Nylon - When knotting long, stone bead necklaces, nylon works very well. Nylon can also be purchased on long rolls or on cards with attached needles. Since nylon is a synthetic fiber, it doesn't stretch or fray. I like the way nylon makes stone bead necklaces drape, even if you're not knotting between the beads. Nylon also comes in different colors and sizes. Most often, I use size #4. It seems to work well with 6mm and 4mm beads, which I use a lot. For small beads, I use size #2.
Monofilament - I've seen this in craft stores labeled as "Jewelry Thread." But, yes, it's that same stuff they use on a fishing pole. When I first started string beads, I used 20 lbs. test filament for everything. Today, I'm a lot more experienced and, therefore, a lot more particular. However, I still think this is okay to use for those cheapo $3 bracelets that are sold as impulse items. Most people who buy a $3 bracelet are going to wear it for a few months and then get tired of it. However, I would never string beads that could cut or stretch the monofilament such as hematite or crystals. Even a $3 bracelet should not break too easily. Though I've never had anything break using filament, and I've got some old stuff I made 10 years ago using it that I still wear sometimes, the cord will eventually become oddly shaped if stored twisted up. So, use your own judgment. If you're making a bunch of stuff to stick in your bargain bin you might want to use it. However, if you're making necklace you're going to sell in a gallery, I would consider another type of cord.
Nymo - This is a staple item in most seed beaders' bead boxes. It comes in a good range of colors and various sizes. You can buy it on large rolls or tiny bobbin sized rolls. Most often, beaders use the smaller rolls for portability and so they can have more colors. It must be waxed using either bees wax or a product call Thread Heaven. Though I seed bead, I'm not into nymo after being introduced to slamide. Nymo is used mostly for seed beads, but can also be used with pearls or heishi.
Silamide - This thread is also used with seed beads. However, it is pre-waxed with strands twisted together, so it's convenient and strong. Originally, many seed beaders weren't very impressed because the color selection was very limited. However, it now is available in all kinds of colors like pink, mustard, turquoise, etc. Another reason some seed beaders don't like it is because it can be difficult to thread through a needle since it is twisted.
Coated Wire aka Beading Wire - Probably some of the best products in recent development for bead stringing are the various types of coated wire threads now available. You've probably heard of Soft Touch, Beadalon, Accuflex, and Soft Flex, which are trade names for this type of material. Depending on the manufacturer, there are various sizes and colors available. This cord works great with crystals, all types of stone beads, and even the thinner sizes can be used with some pearls. The smaller sized cords can be knotted on the end to be used with bead tips (clam shells), but they are most often finished off with crimp beads. This cords is also great to use when making illusion necklaces.