Kala In-Ze was the chief physician of Argo City, and the mother of Kara In-Ze, also known as Supergirl.
When Argo was torn from its orbit by the destruction of Krypton, the surviving Argoans slowly froze to death. When the survivors' distress calls went unanswered, Kala, the planet's chief physician, placed herself and her family in cryonic stasis, hoping that someone would find and rescue them in time.
Before placing herself in stasis, Kala recorded a message for whoever found their stasis chamber, sketching the history of Argo's fall and ending with a plea for help.
Unfortunately, the expanding ice sheets of Argo encrusted the building, eventually causing icicles to fall on the chambers, damaging three of them and causing the occupants to die from exposure to the elements. The only undamaged chamber was Kara's. She was found by Superman, who responded to Kala's distress signal while exploring the system where Krypton had been.
When she was brought to Earth, Kara told Clark that she had dreamed of saving her mother's life and helping her lost friends, and that was why she wanted to emulate Clark in using her powers to become a super-heroine.
Kala's background is explained in Superman Adventures #21. Born Kala Im-Re, she married Zor In-Ze, a physicist, and had three daughters of whom Kara was the eldest. Kala's husband and younger daughters perished in the aftermath of the destruction of Krypton which started Argo's chain reaction (they had been doing errands in Argo's capital city, which was leveled). At the time of the cataclysm, Kala had been planning a dinner party and her brother and nephew had arrived shortly before this happened (the explosion that killed Zor also killed her brother's wife). Scenes are also shown as Argo slowly transforms from a warm green world into a blue ice planet, and the family scavenging for supplies while evading surviving Argoans who have resorted to outright violence to survive. They end up as the four placed in stasis. Superman later reveals to Kara that unfortunately, falling ice damaged the capsules of her mother, uncle and cousin.