John Who

John and Gillian were a young brother and sister and the grandchildren of the Doctor, with whom they traveled as his companions.

Biography
In 1960s England, John and Gillian were looking for their grandfather, aware that he was some kind of inventor, when they found the Doctor's TARDIS in an empty yard. When they entered, 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 travelled with the Doctor for many adventures, fighting many 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 John and Gillian, but reunited with them while in his second incarnation (although, from their perspective, little time had actually passed). They resumed their adventures with him 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 Col. Marc Forest of Special Space Security. (DWCC: Beware the Trods!)

During a visit to the Land of Fiction, the Doctor encountered fictional duplicates of John and Gillian, which had been created by the new Master of the Land, Jason. (NA: Conundrum)

The Doctor later had a dream in which he had an adventure with John and Gillian. (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 prior to meeting the Doctor. MA: Cold Fusion revealed that the Doctor's children were culled but that Susan was rescued, so his other grandchildren may have likewise been saved. It is also unknown if they are brother and sister to Susan, or, if they are her cousins. If so, then the Doctor had at least two children, and, since DWCC: Beware the Trods! confirms that John's surname is Who, their father is likely to be the Doctor's son.
 * 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. Comic strips, particularly those produced before Marvel Comics obtained the license to Doctor Who and began publishing stories with closer ties to the TV series, exist in a "grey area" of canon, neither officially canonical, nor non-canonical. In other words, to date there has never been anything broadcast that either confirms nor denies that John and Gillian weren't additional grandchildren of the Doctor, or that they didn't travel with his first two incarnations at some point in his history. Exactly when is a matter of some debate, as John and Gillian have accompanied the Doctor in adventures in which he meets creatures he'd previously met in the TV series, yet particularly in the case of the First Doctor, it's difficult to rationalize a point in his post-An Unearthly Child timeline in which he would have travelled without any of his TV companions, however, he is alone in The Five Doctors until he is reunited with Susan, so, it is possible that he did have a bit of a gap. As for the Second Doctor, it's possible his reunion with John and Gillian occurred during the Season 6B continuity.
 * Although DWM: The Land of Happy Endings and NA: Conundrum imply that John and Gillian only existed in fiction or in dreams, other interpretations are possible, such as that Jason copied existing individuals, or that the Doctor was remembering a simpler time.
 * At the conclusion of TVC: The Monsters from the Past, Gillian shoots the villain to death, making her one of the only non-UNIT recurring characters (other than Leela) to be shown killing. Interestingly, another female Time Lord, Romana, would later be shown using a firearm in similar fashion in DW: The Pirate Planet.
 * The fates of John and Gillian during and after the Last Great Time War are unknown, but the Doctor believed that all Time Lords died in the conflict.