Alpha Eridani II was an inhabited planet. It was the second planet of the Alpha Eridani planetary system. This planet was aligned directly between Earth and Argelius II. By 2156, this world had been colonized by Humans.
In the real world, the star Alpha Eridani (or "Achernar") is located approximately 139 light years from the Sol system. According to Captain Jonathan Archer, in ENT: "Two Days and Two Nights", set in 2151, the furthest that Humans had traveled at that point was 90 light years. This suggests the colony was established some time between 2151 and 2156. Heliopolis ("Sun" + "City") was a Greek word from antiquity, and was the name of a city in ancient Egypt.
Star Trek Maps, Star Trek Concordance and an entry on Startrek.com refer to the planet as "Alpha Proxima II". The pronunciation of this planet in the episode sounded more like "Alpha Aradoni II", which was also supported by the closed captioning. The spelling of "Alpha Eridani II" can be found in the Redjac entry on Startrek.com and in the Star Trek Encyclopedia and Star Trek: Star Charts.
According to Star Trek Maps (pg. 22), Alpha Proxima II was one of the first extra-solar Human colonies, with major industries in mining, farming and sea-farming. By the 23rd century, the planet had a total population of 12 million with more than 3 million living in the port city Heliopolis. According to the Star Trek: Star Charts ("United Federation of Planets II"), Alpha Eridani (Achernar) was a binary star system. Alpha Eridani A and B were both B-class stars with a magnitude of +1, which was a hundred times brighter than Sol.