Argo City was one of three principal city-states located on the continent of Lurvan on the planet, Krypton (the other two were Kandor and Kryptonopolis). When a great cataclysm shook the planet in 1938, a large portion of the terrain surrounding Argo City heaved itself from the planet's surface, and actually broke through the atmosphere just as Krypton ultimately exploded. The city section maintained a portion of Krypton's gravity and atmosphere, despite the fact that it no longer revolved around the star, Rao.
For the next fifteen years, Argo City drifted harmlessly through space. In time though, the geology of the island became unstable and deposits of Kryptonite broke through the planetoid's surface. A scientist named Zor-El developed a means to stabilize the environment and to safeguard Argo City's populace from the deadly Kryptonite radiation. By covering the entire environment with a specially designed lead-based metallic plating, Zor-El managed to keep his fellow Kryptonians safe for many years.
One day however, Argo City passed through a meteor storm. The meteors rained down on the city, pounding through the protective lead shielding. Kryptonite radiation washed over Argo City killing hundreds of civilians. Fearing that their own deaths were imminent, Zor-El and his wife, Alura placed their sole child, Kara inside of a small rocket ship and exiled her from Argo City. Supergirl's rocket eventually arrived on the planet, Earth, where she ultimately met with her own Kryptonian cousin, Superman.
Zor-El and Alura meanwhile, survived the destruction of Argo City through an alternate means. Zor-El had discovered a twilight dimension he called the Survival Zone. In the city's final moments, Alura and he used a special projector to transport themselves into this zone, where they remained for many years. They reunited with Kara Zor-El when the latter had grown into adult hood and relocated to the city of Kandor on Rokyn.
Years after Argo City's destruction, a race of aliens known as the Vrangs (the original conqueror race of Krypton) discovered the ruins of Argo City floating through space. They examined a stone monolith inscribed with the words, "Whoever finds this monolith, know that you stand before the final bastion of the obliterated planet Krypton... [sic] and the final resting place for the souls of the last survivors of the noble Kryptonian race! Let it be known that we called our home Argo City... and that its citizens died with dignity!" 
In Post-Crisis continuity, Argo City was the home of Zor-El, his wife Alura and their teenage daughter, Kara. Unlike many on the Kryptonian Science Council, Zor-El heeded his brother Jor-El's warnings that the planet would soon explode. Zor-El reverse-engineered force field technology left behind from when the alien Coluan Brainiac, conquered and stole the city-state of Kandor. Although Zor-El could not perfect the device, he did succeed in creating a force field bubble that enabled Argo City to survive Krypton's destruction.
Argo City drifted through outer space as a lonely planetoid for many years. Brainiac eventually sensed that his technology was being used elsewhere and tracked Argo City down. He assimilated the city and merged it with the bottled city of Kandor. Zor-El and Alura were able to send their daughter away in a rocket ship, thus avoiding being captured by Brainiac. 
Now a part of Kandor, Argo City remained in Brainiac's ship for many years until its populace were finally rescued by Superman.  The merged cities were regrown to proper size, and were temporarily placed in the frozen tundras of the North Pole, not far from the Fortress of Solitude. Argo City and Kandor were soon re-dubbed "New Krypton" and was transplanted a second time, this time to a planetoid in direct opposition to Earth's orbit.