Elysium was a name applied to possibly two places — or one. One account referred to Elysium as a location where people could swim with the Obi-Amphibians (PROSE: Sky Pirates!) and another account referred to Elysium as being a lost Earth colony. (PROSE: Parasite)
The latter account also referred to the Elysium system as being a "lost colony of distant Earth" and the site of the Artifact. (PROSE: Parasite)
Earth colony
Elysium was an Earth colony. It was isolated from Earth and the Earth Empire for centuries. It had at least one, likely several, oceans that produced severe tidal events. Elysium was first colonised when the colony ship crashed.
Civil war
When the Seventh Doctor, Ace, and Bernice Summerfield encountered individuals from Elysium on the Artifact, the planet was on the brink of civil war.
The "Founding Families" ruled Elysium. They married each other and stayed away from the general population. They favoured an isolationist system with a religion promoting many children and the sanctity and protection of life.
On the other hand, the Reunionists, who had arisen after the discovery of the Artifact, supported attempting to contact the Earth Empire and breaking out of the closed system of Elysium. (PROSE: Parasite)