Nylon is an artificial fiber, invented in 1935 by a bright chap at DuPont. It was intended to be an replacement for silk, but quickly found many other applications. Like many other artificial fibers, it’s a polymer, usually derived from petroleum. It’s extruded as a very fine, very smooth strand. How smooth? This is what it looks like when magnified forty times.
In clothing, “nylons” are synonymous with tights/stockings, given one of their more popular applications shortly after introduction.
Nylon fabric is durable, easy to care for and fire resistant (although that means it’s going to melt on your skin rather than burst into flames – scant consolation). When it was first introduced, it was hailed as a wonder fabric, especially for women working as a part of the war effort who wanted dresses that were easier to care for and longer-lasting than natural fibers.
It’s unlikely that you’re going to encounter a nylon fabric – or even a blend – at the fabric store, as it has generally been supplanted by better alternatives. The most likely exception to this are the “ripstop” fabrics you might find in the outdoor/utility section. Ripstop can be used by a cosplayer for making durable bags, banners and waterproof outerwear.
Some nylon sheers are out there, usually wide-mesh netting, but you’ll find a picture of nylon chiffon in the depths of this website, too.
If you have to wash it, hand wash it in lukewarm water and let it drip-dry. Some nylons are safe in the dryer on low or medium heat – check the bolt when buying.