Superman — The Man of Steel himself is probably the best incarnation of the super family. He was the originator of the series and thus carries the name on when other incarnations drop in and out of the DC continuity, like Supergirl. The others in the lineup are literally just watching the throne.
Superboy — There have been several versions of Superboy but the most prominent is the little boy who would become Superman. DC has since stated that Superman didn’t have adventures until he became an adult but that hasn’t stopped the multitude of other versions, such as the post-Crisis on Infinite Earths being.
Supergirl — Supposedly, the story goes that the daughter of Zor-El — Superman’s uncle — survived the explosion of Krypton since she was living in Argo City, which was cast off into space when the planet exploded. Kara Zor-El was the last survivor, giving Superman one of his only living clansmen known to have survived the catastrophic event that created the lore. Her existence is removed during and restored after the 1985 arc Crisis on Infinite Earths.
Krypto the Superdog — Superman might have godlike powers on Earth, but on Krypton he had decidedly human emotions and that extended to having a pet. Krypto was used as a test subject to experiment with rocket flight. Jor-El — Superman’s father — realized Krypton was going to explode and wanted to test a way to get himself and his family off the doomed planet. Krypto was sent out into space, but the rocket was knocked off course. Drifting through space for years, the dog was found and rejoined Superman during his Superboy years.
Bizarro — The ultimate in “the evil clone/twin did it” storyline, Bizarro exists only because Lex Luthor, as usual, was messing around with things he didn’t understand. Luthor recreated the duplicating ray that was used previously on Krypton by Gen. Dru-Zod and on Earth to accidentally create a duplicate version of Superboy. Luthor also creates a version of Bizarro after the Crisis on Infinite Earths arc while trying to create a clone of Superman.