Pre-CRISIS ON INFINITE EARTHS
Though her original introduction came after the one-shot introduction of three different "Super-Girl" characters in the Golden Age, it was Kara Zor-El who stuck.
Kara's first origin story was similar to what would end up being her modern day incarnation. She was the cousin of Kal-El and the last survivor of Krypton's Argo City, a fragment of the planet that had managed to hold together after the original cataclysm -- at least, for a time. When Argo was threatened by a meteor shower, her family launched the adolescent Kara to safety in a rocket with her destination set to Earth. Her parents planned for her to be found and raised by her cousin, so they dressed her in a uniform similar to his in hopes that it would make her easier to recognize as one of their own.
Once on Earth, Kara adopted the civilian identity of Linda Lee and came to live in Midvale Orphanage. From there, following in her cousin's footsteps of maintaining a cover identity and did her best to keep her powers a secret.
Later, Kara was adopted by a human family, the Danvers (Action Comics #285, 1968) and finally began acting as Supergirl officially, introduced to the world by Superman himself. From that point on, Kara became a fixture of the Superman Family, sharing stories alongside characters like Lois Lane and Jimmy Olsen.
CRISIS ON INFINITE EARTHS
Kara played a featured role in the massive continuity shifting efforts of 1985-86's CRISIS ON INFINITE EARTHS, a 12-issue series that served to condense the DC multiverse into one single universe.
When existence was threatened by an extra-dimensional being called the Anti-Monitor, his equal and opposite figure, the Monitor, began calling upon the DC Super Heroes from different Earths across the multiverse to stop him. Traveling from dimension to dimension, the chosen heroes worked to stave off the Anti-Monitor's attacks, but try as they might, Earth after Earth fell to nothingness.
During one of their last stand efforts in assaulting the Anti-Monitor's fortress, Kara was killed, creating the now iconic cover of her cousin holding her body and weeping as the cosmic scene plays out behind them.
Kara's death served to help realign the Superman Family from their disparate and sometimes confusing Golden and Silver Age roots where "imaginary stories" and alternate realities allowed different variations of Kryptonian characters to pop up in and out of stories at will, making Kal's origin as the last surviving member of his race murky and hard to follow.
Matrix & Linda Danvers
Several years after the death of Kara Zor-El, a new Supergirl entered the scene. Created by an alternate reality Lex Luthor who had not become a villain, this Supergirl was actually an artificial organism known as Matrix, who could shape shift and adapt metahuman abilities at will. Originally, Lex patterned Matrix's memories and physiology after the alternate reality Lana Lang he had been in love with, but soon, when their reality was invaded by three super-villains -- General Zod, Faora, and Quex-Ul -- Lex attempted to re-configure Matrix into a Super Hero.
Replicating a version of Kara -- including blonde hair, blue uniform, and Kryptonian powers -- Matrix became Supergirl and attempted to save her home from the invasion. Unfortunately, however, she was unsuccessful, and in a last ditch effort to save her, Lex sent her to live in the core DC universe.
Matrix came to live with the Kent family when she arrived in the main DCU, and they raised her to the best of their abilities. Giving her the human name "Mae," Matrix struggled with her own identity and natural inclination to adapt and transform to better adhere to her surroundings.
Matrix eventually fell in love with a disguised Lex Luthor, which drove a wedge between her and the rest of the Kent family. It was Lex's intervention that prevented Matrix from coming to help Clark in the fight against Doomsday that would eventually kill him (in 1992's "Death of Superman" storyline), though she did continue to attempt to protect Metropolis in his absence.
Later, Lex's treachery was revealed when Matrix uncovered that he had been secretly cloning her in order to make an army of his own personal "Supergirl" soldiers. She promptly flew into a rage, destroying his lab and the clones. This betrayal sent her into an existential crisis where she began to question whether or not she had a soul. In her quest to uncover answers, she encountered a young dying girl named Linda Danvers (no relation to the Linda Lee Danvers identity Kara Zor-El had used in the Silver Age). Using her shape-shifting and mind-melding abilities, Matrix merged with Linda and the two became a singular entity, the Earth Angel of Fire.
However, this new status was only temporary. Eventually, the Matrix aspect of the Linda/Matrix fusion was absorbed away leaving Linda alone but still semi-empowered. She retained some of her metahuman abilities and continued to serve as Supergirl for a brief period of time.
The third (or, technically, fourth) iteration of Supergirl was Cir-El, the time traveling daughter of Clark Kent and Lois Lane brought to the present by an organization called the Futuresmiths. Though the true nature of her parentage remained up for debate after some tests, Clark came to recognize Cir-El as family and a worthy holder of the Supergirl mantle.
Cir-El worked closely alongside both the Superman and Batman families for several years before the truth of her history came to light. It was revealed that she was not Clark's biological daughter but instead a Kyrptonian/human hybrid named Mia who deeply resented her Cir-El identity and Superman himself.
Pulled between her two warring identities, Cir-El's story came to a close when the Futuresmiths finally came to collect. Caught in an endless, time-skipping war with the villain Brainiac, Cir-El was forced to leap into an interdimensional portal, sacrificing herself, to end the cycle of time loops and dystopian futures her existence kept causing. However, due to her anomalous temporal qualities, Cir-El was still able to return, occasionally, to the present day.
The Return of Kara Zor-El
The original Supergirl was reintroduced to main continuity for the first time following her death in CRISIS ON INFINITE EARTHS for the first time in 2004's BATMAN/SUPERMAN "Supergirl" storyline. This version of the story included most of her original origin story, with the added element of the intention for her to be Clark's caretaker, whose escape pod had been suspended and delayed.
In this continuity, Kara arrived for the first time during the Modern Age, a young and disoriented girl who could only speak Kryptonian discovered upon her landing by both Superman and Batman. Confused and baffled by her appearance at first, both Clark and Bruce were wary to trust her identity, but slowly opened up to the possibility that her story was true upon running a battery of tests.
Throughout the Modern Age, Kara sets out as Supergirl to meet and become more accustomed to the DC Universe and its various Super Heroes and teams.
During the aftermath of 2006's INFINITE CRISIS, Kara was transported to the 31st Century via Zeta Beam, and she joined the Legion of Super-Heroes. She did not return to the present day until 2006's "One Year Later" time-skip event, in which she took over the protection of Metropolis in Superman's absence, alongside Booster Gold.
This version of Kara took on the civilian name Linda Lang, adopting an identity as Lana Lang's niece after she decided to scale back her activities as a Super Hero temporarily. Eventually, the events of 2008's FINAL CRISIS prompted her to become fully active once more as Supergirl, especially in the wake of Batman's "death." She teamed up with Superman in an effort to maintain order in the world as the last of Darkseid's takeover was dealt with (in 2009's SUPERMAN/SUPERGIRL: MAELSTROM).
Later, Kara and Clark discovered the miniaturized, bottled city of Kandor in the ship of Vril Dox, the original Brainiac. Upon defeating Brainiac, Superman was able to restore the citizens of Kandor to their full size and, in doing so, reunited Kara with her parents who had been trapped inside. But the sudden influx of Kryptonians on Earth prompted an uncomfortable military response.
After tensions boiled over, all Kryptonians were relocated to a planetoid in orbit near Earth that came to be called New Krypton. The caveat? None of the Kryptonians who lived there could set foot on Earth again -- Kara included.
This sanction didn't hold for long. Kara returned to Earth in time for the events of 2010's BLACKEST NIGHT, which prompted her to partner up with both Stephanie Brown, the newest Batgirl, and Damian Wayne, the newest Robin.
Kara returned to New Krypton as it was attacked by Brainiac alongside her fellow Superman Family members. Unfortunately, however, they were not able to save the planet and, in the chaos, a newly returned General Zod declared war on Earth (in 2010's "Last Stand of New Krypton" storyline).
After Zod's defeat (and subsequent destruction of the majority of New Krypton's former inhabitants) Kara found herself adrift and in mourning for her people, but her melancholy was interrupted by the sudden appearance of Bizarro-Supergirl. Despite her wishes to retire from the Supergirl identity altogether, Kara was forced to step up and fight her Bizarro doppelganger -- a fight that takes Kara all the way to the Bizarro World itself.
Kara continued to act as a Super Hero across both time and space, temporarily making returns to the Legion of the Super-Heroes in the future and teaming up with her fellow Earthbound heroes in the present until the DC Universe was reimagined following the events of 2011's FLASHPOINT.
The New 52
Post-FLASHPOINT, Kara's origin remained largely the same, with a few minor changes. Her escape pod landed in Russia where she was initially attacked by security robots. When Superman attempted to intervene, Kara believed he was an imposter wearing her family's crest and the two began to fight across the globe before they came to an understanding.
Kara took on the Supergirl mantle and began life as a Super Hero in this new timeline. Eventually, after acting as Super Girl on Earth against various high powered threats, she she inherited a Red Lantern ring after her grief and frustration both at her situation as a Kryptonian survivor, and her inability to save every human in need of help overwhelmed her. She accepted her ring and became a fully fledged member of a Red Lantern splinter group under the tutelage of Guy Gardner. With Guy's help, she maintained her sanity and presence of mind even under the ring's parasitic control.
However, eventually she gave her ring up to return to Earth where she returned to her Supergirl identity until 2016's DC UNIVERSE: REBIRTH #1.
Following DC UNIVERSE: REBIRTH #1, Kara was largely returned to her pre-Flashpoint status quo. She resumed being a teenager living with her adoptive family, the Danvers, in National City. She acts as Supergirl largely to protect her home from cosmic threats, of which there are many. So far, she has faced down Cyborg Superman, Kryptonian werewolves, and members of the Superman revenge squad.