Satyrs are specific to Greek mythology, Fauns are specifically Roman. Satyrs have goat's horns and horse's tails, and usually human legs, but a permanently erect penis. Fauns are more innocent but have goat legs, and are more interested in girls than wine.
The Greek and Roman authors say little about the origin of Satyrs and Fauns. They are minor deities, perhaps descended from Dionysus (Satyrs) or Pan (fauns) with nymphs or dryads as mothers. The ancient sources aren't clear, they didn't see this as important.
Instead the sources focus on them as lovers of wine, women and song. There is a shift, over the classic period, from satyrs as being wild, untamed forest demons, to nature gods, freed from inhibition. The association with music grew over the period.
Perhaps the association with wine and song led to an over-representation since pottery associated with wine has often survived, as have some of the songs poems and plays of the era, in which music plays a large part.
One can't properly speak of satyrs outside of a Greek context. Other cultures may have had goat-legged gods or demons, notably in the Hebrew myths recorded in Leviticus, the saiyr is a hairy or goat-like man and is often translated as satyr, although there is no suggestion that the words are cognate.
