In Greek mythology, the Titaness Styx ('Hated') was the daughter of Okeanos and Tethys. Her offspring were Bia ('Strength'), Kratos ('Power'), Nike ('Victory') and Zelus ('Zeal'). Styx is referred to as a nymph in many mythology sources, and may well have been counted as one by later mythographers.
When Zeus and his Olympian kinfolk rose up to destroy the Titans, Styx aided him in this battle. The Titans were defeated and banished to the gloom of Tarterus (those who weren't simply blown to bits by Zeus' thunderbolts). As a reward for her assistance, the gods thereafter swore oaths by Styx's name. Anyone breaking such a vow would be banished from the company of the gods for nine years.
Styx became the goddess of, or even the embodiment of the mighty river which kept the realms of the living and the dead apart. Grim Charon was the boatman who ferried the souls of the dead to the other side of the River Styx. In order to pay his fare, the dead were frequently buried with a small coin in their mouth or on their eyes. Those who could not pay became a sort of spectral panhandlers, cursed to wander the realm of the living until they could get someone to pay their way into Hades' realm. It was a common enough act of philanthropy for wealthy or powerful people to put up big sums of money at festivals in order to send a big bunch of wayward souls to their reward. I don't know if the boatman accepted them first-come first-ride or if the money was doled out as a sort of ghostly feeding frenzy, like a department store at sale time.
The river (originally supposed to be in the region of Arcadia) and her many tributaries had magical properties. The River Lethe ('Forgetfulness') could erase the memories of any who touched its water. Some believed that the souls of the dead were baptized in the Lethe's water to remove their memories of life. The river Phlegethon ('Burning') burned the wicked in a stream of lava. Baby Achilles was dipped in the Styx's magical water to confer invulnerability, but Thetis, his regal mom, held him by the heel, creating a small spot of vulnerability there.
The idea of an underworld river has precedents. The Epic of Gilgamesh talks about the great hero having to traverse a river, complete with a dark boatman. The Stygian imagery made itself into popular Christian imagination as well. Many writers, including Dante, included a gloomy or fiery river that the dead had to pass over into Hell.
Monaghan, Patricia, "the Book of Goddesses and Heroines" (Llewellyn, St. Paul, MN, 1990).
Graves, Robert, "the Greek Myths" (Penguin, Middlesex, 1955, 1985).
Cotterell, Arthur and Storm, Rachel, "the Ultimate Encyclopedia of Mythology" (Hermes House, London, 1999).