Honestly, when I set out to write this, even though I knew it was not accurate, this is what my mind thinks a satyr looks like:
Or perhaps like this:
What I tend to think of as satyrs are actually fauns. A much better example of a satyr would be this:
In true Greek mythology, satyrs were between a faun and the Narnia satyr above. In Greek mythology, satyrs were closely tied with Dionysus though they were also known to cavort with and serve Gaia, Rheia, Hermes, and Hephaestus. They were most often the companions of Dionysus, drinking and playing flutes or tambourines. The flute was their preferred instrument.
Satyrs had the head of a man, but had pug noses, donkey ears, donkey or horse hind legs, and a horse tail. Though their body was mostly hairless, they were almost always depicted with long dark hair and beards. If they wore clothes, they were made from animal skins with the fur still intact and wore laurels of vines or ivy on their brow. They were considered to be symbols of nature, life, and the harvest, and as such were often shown with large, erect members. *cough* If you do a Google image search for satyr quite a few interesting things come up. I would advise you not trying that one at work.
Satyrs often consorted (sexually) with the nymphs, maenads, and other bacchanals. By some accounts, they were adapt at every kind of sensual pleasure. Their main purpose seems to be to follow Dionysus around, drink, and pursue females of all kinds.
Their parentage is disputed. The most widely held belief is that the satyrs were the sons of Hermes and Iphthima or that they were descended from the Naiads. They were also claimed by Silen. Strabo wrote that they were sons of the five daughters of Hecataeus and the daughter of Phoroneus.