John Who

John and Gillian were siblings. According to some accounts, they were also the grandchildren of the Doctor, who travelled with both the first and second incarnations of the renegade Time Lord. (TVC: The Klepton Parasites — Invasion of the Quarks) According to other accounts, they may not have ever existed, but were instead either the dreams of the Doctor, (DWM: The Land of Happy Endings) or the creations of the Master of the Land of Fiction. (NA: Conundrum)

As real beings
Some accounts held that John and Gillian were real, corporeal beings.

According to these, John and Gillian first met their grandfather in 1960s England. They were aware that he was some kind of inventor, but had evidently never seen him or been travelling with him before, when they found the Doctor's TARDIS in an empty yard. They entered and he acknowledged them as John and Gillian. During their visit, John playfully touched a button on the console and the TARDIS took them to the 30th century, where they helped the Thains defeat the invading Kleptons. (TVC: The Klepton Parasites)

John and Gillian then travelled with the Doctor on many adventures, fighting enemies such as Captain Anastas Thrax, the Zarbi, the Gyros robots, the Pied Piper and the Trods. At some point, the Doctor parted company with them, but reunited with them while in his second incarnation. From their perspective, little time had passed. They resumed their adventures and fought other enemies, including the Daleks and the Cybermen until they reached their teen years, at which point the Doctor enrolled them in a university on Zebadee to keep them safe from the Quarks. (TVC: Invasion of the Quarks)

John eventually became a professor and wrote a memo on the dangers of the Trods to Colonel Marc Forest of Special Space Security. At this time, John styled himself "Professor John Who". (DWCC: Beware the Trods!)

As imaginary beings
Other accounts suggested John and Gillian were imaginary.

On one visit to the Land of Fiction, the Seventh Doctor encountered a "John" and "Gillian" who claimed he was their grandfather, with whom they'd had Klepton- and Trod-fighting adventures. They tried to make him remember that his name was "Dr Who", to which he flatly replied, "I can assure you, I'm not." When they pressed further and asked if he remembered them, he said that he did not. (NA: Conundrum)

The Eighth Doctor later had recurring dreams in which he adventured with John and Gillian. These dreams helped the Time Lord unwind by suggesting a reality in which the solution to problems was uncomplicated and even easy. After one such dream, the Doctor emerged refreshed and re-energised. (DWM: The Land of Happy Endings)

Behind the scenes

 * It does not appear that John and Gillian had met the Doctor before The Klepton Parasites, although they were aware of his existence, and the Doctor's comment upon their arrival implies that he was expecting them. Nothing was ever revealed about their lives before meeting the Doctor.
 * The characters of John and Gillian were created for the Doctor Who comic strip in lieu of using (and paying license fees for) any of the companions from the TV series.
 * At the conclusion of TVC: The Monsters from the Past, Gillian shoots the villain dead, making her one of the only non-UNIT recurring characters to be shown killing.

John y Gillian