Short answers, pulled from the story.
A satyr in Greek mythology is a male nature spirit with the ears and tail of a horse, a bestial face, and a snub nose, always depicted nude. Satyrs were companions of the god Dionysus and were associated with wine, music, dancing, and the pursuit of nymphs and mortal women. They were believed to inhabit remote woodlands, mountains, and pastures.
Classical Greek satyrs had horse-like features and were associated with secret wisdom as well as lust and revelry. Roman fauns were generally depicted as shyer woodland creatures and unambiguously goat-like, with the upper body of a man and the legs, hooves, tail, and horns of a goat. The Romans identified fauns with their own native nature spirits, and eventually the distinction between the two was lost entirely.
Marsyas picked up the aulos, a double-pipe instrument that the goddess Athena had discarded and cursed. He became skilled enough to challenge the god Apollo to a musical contest. Apollo was declared the winner after Marsyas could not invert his instrument and play it upside-down as Apollo had done with his lyre. Apollo then hung Marsyas from a pine tree and flayed him alive as punishment for his hubris. The Athenian sculptor Myron commemorated the story in bronze sculptures installed before the Parthenon around 440 BC.
The only complete surviving satyr play is Cyclops by Euripides. It is a burlesque of a scene from the Odyssey in which Odysseus is captured by the Cyclops Polyphemus. In Euripides's version, a tribe of satyrs led by Silenus has been enslaved by Polyphemus, and Silenus attempts to manipulate both Odysseus and Polyphemus for his own benefit, primarily to obtain wine.
Satyrs originally had horse ears and tails and sometimes horse legs, but by the sixth century BC human legs were more common in art. During the Hellenistic Period, from 323 to 31 BC, their iconography was gradually conflated with that of the Pans, plural forms of the god Pan, who were depicted with the legs and horns of goats. Since the Renaissance, satyrs have been most often represented with goat legs and horns.
In the Middle Ages, Christian writers beginning with Jerome portrayed satyrs as symbols of Satan, and satyrs appeared in bestiaries dressed in animal skins or shown as a species of ape. During the Renaissance, artists gave them both human and goat features in whatever proportion they chose. Albrecht Dürer's 1505 engraving The Satyr's Family depicted them in domestic, familial scenes, a widely reproduced image that may have contributed to the later European idea of the noble savage.