Pre-Crisis Krypto showed up in the old Superboy comics. Jor-El rocketed the dog into space as a test before strapping his only son on a rocket. The more obsolete spaceship arrived on Earth after Superboy's, and it was Superboy's idea to strap a cape onto Krypto (who spent his days chasing meteors). It was shown that while Superboy adventured with the Legion of Super-Heroes, Krypto was a member of the Legion of Super-pets, alongside Streaky the Supercat, Comet the Superhorse and Beppo the Supermonkey.
The came the Superman relaunch, and with it the notion that Kal-El was the sole survivor of Krypton and never was Superboy.
After Superman's "death" at the hands of Doomsday, Cadmus Project mixed kryptonian and human DNA and created Superboy. He eventually was attached to a white mongrel (who was totally normal, not super). The dog was gonna be named Krypton, but the bone-shaped dogtag was too short and the printer left the last "N" out, so the mutt became Krypto. He lived with Superboy in Hawaii for a while and everyone in the DCU seems to have forgotten him.
A couple of years back, Superman and Lois travelled to an alternate Krypton, very similar to the Pre-Crisis one. There they met Jor-El, Lara and the family dog, Krypto. When Supes and Lois returned, Krypto followed. On Earth he received Superman-level powers, but has the intelligence of a regular dog (unlike the supersmart pre-Crisis Krypto). When Mongul attempted to attack Lois, Krypto (like any dog) went for the throat and punctured it with his Superman-level strength. Since then, Superman has kept Krypto in the Fortress of Solitude.
Recently, Superman asked Superboy to train Krypto, so the super-mutt became sort of Superboy's dog. It seems Superboy doesn't even remember his first dog.