Born Kara Zor-L of Earth-2, Power Girl escaped the destruction of Krypton when she was a child via a rocket ship in which her parents placed her. Although she left at the same time her cousin, Kal-L, did, her ship took much longer to reach Earth-2. Once there, Kara discovered she had superhuman powers and became a protector of humanity. She behaves much older, more mature, and more level-headed than her Earth-0 counterpart, Supergirl. She was also a member of this world's super-heroic team, called the Justice Society of America, and her best friend is Helena Kyle, aka the Huntress.
And then came the Crisis. Power Girl survived, but Earth-2 was no more. She could still recall her true past, but eventually the multiverse adjusted her, and she came to believe she was the descendant of the Atlantean sorcerer called Arion and had been frozen in suspended animation for millennia until the present day. Once awakened, a spell Arion had placed upon her made her think she was Superman's cousin. At first, Kara was even much weaker than she originally had been, but over time she became just as strong as she used to be.
And then came ZERO HOUR. Her origin as the granddaughter of Arion was revealed to have been a lie concocted by her adoptive mother.
And then came INFINITE CRISIS. The evil Psycho-Pirate took advantage of the reality changes mucking about with the universe at the time and manipulated Power Girl into believing she was born a Daxamite called Laurel Gand in the 31st Century, sent back to the present day as a preemptive measure against the sorcerer called Mordru. He later manipulates her into believing she was actually from Qward and was the cousin of that world's Ultraman; the daughter of Conner Kent and Cassandra Sandsmark of the Titans Tomorrow reality; the daughter of Shazam; and the daughter of Ursa (born in the Phantom Zome). However, Power Girl ultimately remembers her true past thanks to the subconscious mental tether that connects her to the original Earth-2.
And then came FLASHPOINT. Power Girl is once again from Earth-2 and cousin to Superman. However, this time, Superman and his wife, Lois, adopted Kara when she was younger and raised her to adulthood. Although he is her cousin, Power Girl thinks of him more as a father figure, with Lois as her mother. He trained Kara to be a secret weapon against the coming of Darkseid, dubbing her Supergirl. Her best friend is once again Helena Kyle, aka Huntress, daughter of this Earth-2's Batman and Catwoman, like before. In addition, Power Girl has no knowledge of her history prior to FLASHPOINT, but this could be only a matter of time.
SIDE NOTE: It's possible the current Power Girl of Earth-2 is NOT the original remade. Another Power Girl appeared in the previous HARLEY QUINN series, where she was amnesiac, having fallen out of the sky without warning. This Power Girl was more reminiscent of pre-Flashpoint Power Girl. She eventually regained her memories back, but to what extent remains unknown. However, it is more likely this was New 52 Power Girl, having been knocked back to Earth-0 due to events in the EARTH 2: WORLD'S END series. But then, how did she return to the Earth-2 universe after this? Unless for Power Girl, this took place BEFORE her return to Earth-2 in WORLDS' FINEST #26. It's hard to reconcile to be sure, as HARLEY QUINN is a series that skirts around issues of continuity.