In ancient geography there occur several rivers of this name, all of which were, at least at one time, believed to be connected with the lower world.
The river first looked upon in this light was the Acheron in Thesprotia, in Epirus, a country which appeared to the earliest Greeks as the end of the world in the west, and the locality of the river led them to the belief that it was the entrance into the lower world.
When subsequently Epirus and the countries beyond the sea became better known, the Acheron or the entrance to the lower world was transferred to other more distant parts, and at last the Acheron was placed in the lower world itself. Thus we find in the Homeric poems (Od. x. 513) the Acheron described as a river of Hades, into which the Pyriphlegeton and Cocytus are said to flow. Virgil describes it as the principal river of Tartarus, from which the Styx and Cocytus sprang.
According to later traditions, Acheron had been a son of Helios and Gaea or Demeter, and was changed into the river bearing his name in the lower world, because he had refreshed the Titans with drink during their contest with Zeus.
They further state that Ascalaphus was a son of Acheron and Orphne or Gorgyra.
In late writers the name Acheron is used in a general sense to designate the whole of the lower world. The Etruscans too were acquainted with the worship of Acheron (Acheruns) from very early times, as we must infer from their Acheruntici libri, which among various other things treated on the deification of the souls, and on the sacrifices (Acheruntia sacra) by which this was to be effected.
This is a description of the Acheron and the lower world in general from Plato's Phaedo:
The hollows on the surface of the globe vary in size and shape from that which we inhabit: but all are connected by passages and perforations in the interior of the earth. And there is one huge chasm or opening called Tartarus, into which streams of fire and water and liquid mud are ever flowing; of these small portions find their way to the surface and form seas and rivers and volcanoes. There is a perpetual inhalation and exhalation of the air rising and falling as the waters pass into the depths of the earth and return again, in their course forming lakes and rivers, but never descending below the centre of the earth; for on either side the rivers flowing either way are stopped by a precipice. These rivers are many and mighty, and there are four principal ones, Oceanus, Acheron, Pyriphlegethon, and Cocytus.Translated by Benjamin Jowett
Oceanus is the river which encircles the earth; Acheron takes an opposite direction, and after flowing under the earth through desert places, at last reaches the Acherusian lake,--this is the river at which the souls of the dead await their return to earth. Pyriphlegethon is a stream of fire, which coils round the earth and flows into the depths of Tartarus. The fourth river, Cocytus, is that which is called by the poets the Stygian river, and passes into and forms the lake Styx, from the waters of which it gains new and strange powers. This river, too, falls into Tartarus.