Acheron was an ancient kingdom, extinct in Conan's time. It was founded by a northern offshoot of Lemurian survivors several thousand years after the Cataclysm.
Acheron covered an area roughly corresponding to Aquilonia, Argos, and Nemedia, with her dominion later extended over western Koth, Ophir, and Corinthia. The last three nations were under the control of Old Stygia, before being assimilated into Acheron at a later date. It's likely that Acheron overextended itself by annexing these Stygian satellites, leaving its northern frontier vulnerable to the subsequent Hyborian invasions.
Little is known about the chronology of Acheron; but it seems to have flourished for about 2,500 years as a theocracy under the control of necromancers and wizards. The barbarian pressure of the more vigorous Hybori came down from the north and ultimately destroyed Acheron. Its demise took place 3,000 years before the time of Conan.
The wizard Xaltotun attempted to restore the ancient kingdom, but was stopped by Hadrathus, a priest of Asura, and Zelata, a witch, with power over the Heart of Ahriman. According to a Conan tale featured in Marvel Comics, the Hyborean Age is littered with remnants of the Archeron Empire, from relics to lost cities.