Argelius II was the inhabited second planet of the Argelius system. This system was located approximately 115 to 120 light years from the stellar cluster Jewel stars. The planet was the homeworld of the Argelians, a humanoid species. (TOS: "Wolf in the Fold"; TNG: "Datalore", "Conspiracy", production art)
Two hundred years before 2268, the Argelians experienced what they call the "Great Awakening". In this awakening, they united under an uniglobal government and created a legal system that was based on love. This government was headed by a Prefect. The seat of government was located in a chief city. Due to inefficiencies in their government, people from other planets were hired to fill certain administrative positions.
Prior to stardate 3619, Argelius II signed a treaty with the Federation wherein Starfleet could use the planet as a space port for their starships and crews from the ships could take shore leave on the planet. For male crew members, the Argelian women were considered one of the highlights.
In 2267, the Prefect was Jaris. In that year, the planet was the scene of several murders committed by the entity known as Redjac. Captain Kirk and Lieutenant Commander Scott both enjoyed several visits on the planet, though Scott was framed for the murders during his visit. (TOS: "Wolf in the Fold")
In 2364, the location of Argelius II was labeled on a star chart filed in the USS Enterprise-D library computer. The planet Alpha Eridani II was on a direct alignment between Argelius II and Earth. (TOS: "Wolf in the Fold"; TNG: "Datalore", "Conspiracy", production art)
"ar-GEE-lee-us" was the pronunciation for this planet's name from the script pronunciation guide for "Relics". 
According to the The Worlds of the Federation, the star that Argelius II orbits is named Argelius B (apparently part of a binary star system with Argelius A), and the Argelians refer to their world as Nelphia. According to Star Trek: Star Charts ("United Federation of Planets I & III"), Argelius II was also known as Iota Leonis II. In 2378, Argelius II was a Federation member. The Argelius System, also known as the Iota Leonis System, was a binary star system. Argelius A was a F-class star with a magnitude of +7, which was 1/10 as bright as Sol. Argelius B was a F-class star. The system was located in or near Federation space.
The star chart seen in "Datalore" and "Conspiracy" originally appeared on page 133 of the Star Trek Spaceflight Chronology, where it was explained as showing a part of Federation space in the 2140s, with the thin lines denoting routes of commercial tourist starliners.
According to the novel The Sorrows of Empire, the mirror universe Montgomery Scott was likewise accused of brutally murdering three women on Argelius II. However, Captain James T. Kirk ensured that the charges were dropped without an investigation taking place.