When Supergirl first appeared, Kara Zor-El was an orphan that went by the name Linda Lee on Earth. After a few years, however, she was adopted by Fred and Edna Danvers to become Linda Danvers. That was the status quo until Supergirl died in Crisis in Infinite Earths.
Post-Crisis, Supergirl was a [long story] named Matrix. But when she got her own series in 1996, part of the premise was that she bonded with a teenage girl named Linda Danvers, and could switch back and forth between them. Linda's parents were Fred and Sylvia Danvers.
Eventually, a Kara Zor-El version of Supergirl arrived post-Crisis. She lasted into the New 52, but neither of these versions had adoptive parents (well, at first; more on this in a bit).
Last year (2015), Supergirl got her own tv series. In it, she was adopted by Jeremiah and Eliza Danvers and started going by Kara Danvers. As Supergirl, she would work for the Department of Extranormal Operations, and over the course of the first season, we would learn that Jeremiah did as well.
Supergirl Rebirth #1 (August 2016) continues the story of the New 52 Superman, but changes it to match the show as much as it can. Supergirl is now working for the DEO in National City (where she lives in the show), which is led by Cameron Chase (who also appears in the show) instead of Director Bones. She is given two DEO agents, Jeremiah and Eliza Danvers, to be her handlers and "parents". Therefore, Supergirl now has Kara Danvers as a secret identity, and has the same brown hair/ponytail/glasses look she does on the show. It's not exactly adoption, but maybe it'll be made legit down the road.
And no word yet on whether they have a daughter named Alex, but I'm sure she'll show up sooner than later.