After Krypton explodes Argo is flung into the far reaches of space and the ground slowly turns to Kryptonite. Thanks to Zor-El's quick thinking the citizens line the floor with lead plating sealing in the Kryptonite radiation. The people lived happily and Zor-El bore a daughter Kara Zor-El. Years later Argo City is hit by a meteor storm which pierces the lead shielding exposing the residents to Kryptonite. Zor-El sends his daughter to Earth in a rocket the same way his brother had done years before with Kal-El. The city dies.
After Brainiac sets Krypton's star 'Rao' to supernova Krypton explodes. Zor-El uses technology left by Brainiac when he stole Kandor to form a dome to protect Argo City from Krypton's explosion. The people of Argo live safely for a few years until one of Brainiac's drones alerts Brainiac of Argo's survival. Acting as a homing beacon it draws Brainiac back. Kara is sent to earth in a rocket ship which is compromised by Brainiac as it tries to leave. This allows kryptonite radiation to seep in as she sleeps on her journey. Brainiac takes Argo and adds it to the Bottle City of Kandor, killing any unnecessary information. The bottle city is stored in Brainiac's ship until its discovery and enlargement by Superman.
In the 'Supergirl' feature film Argo is in an alternate dimension. They live in a hippy commune-like culture. The city is powered by an Omegahedron which is lost at the start of the film. Kara must find the power source before Argo City dies. While Kara is said to be Superman's cousin there is no suggestion Argo was ever part of Krypton.
DC Animated Universe
In the DCAU Argo was the sister planet of Krypton that was shunted out of orbit when Krypton exploded. Argo became a frozen landscape and its society descended into violence (as shown in the Supergirl Adventures comic one-shot). Alura decides to put Kara, herself and two other family members in cryogenic tombs until they are rescued. A cave-in destroyed all but Kara's tube and she was rescued by Superman.