The Doctor's mother

According to several accounts, the Doctor remembered having a mother.

The Eighth Doctor claimed that his mother was human; (TV: Doctor Who, PROSE: Alien Bodies, The Infinity Doctors, Grimm Reality, Unnatural History, The Shadows of Avalon) she was, at any rate, a Time Lord. (COMIC: The Comfort of the Good) The Eighth Doctor also said that he couldn't remember if he had a mother or if he was loomed. (PROSE: The Shadows of Avalon, Bafflement and Devotion) If the Doctor's mother did exist, the Thirteenth Doctor believed her and the rest of her family to be dead. (TV: The Woman Who Fell to Earth)

Biography
The Eighth Doctor, when being interrogated about his origins, once claimed that he was "from Andromeda" and that "[his] mother had been abducted by little green men". (PROSE: Seeing I) The Doctor had earlier confessed to being half-human "on his mother's side". (TV: Doctor Who)

The Doctor said that his mother used to sing the Zagreus nursery rhyme to him. (AUDIO: Seasons of Fear) When the Doctor was just a "small child", the Doctor's mother sat by the side of his bed and told him the story of Grandfather Paradox. (PROSE: The Gallifrey Chronicles)

The Eighth Doctor relived a memory from his first incarnation where his mother watched as his father held him up as a child to see the stars. (PROSE: The Eight Doctors) The Doctor remembered that her name was Penelope. (PROSE: The Infinity Doctors)

Ultimately, the Doctor chose to think of the universe as his foster family, after his parents "decided to opt out of their responsibilities." (PROSE: Beltempest)

The Thirteenth Doctor stated that her entire family had died a long time prior to her regeneration. (TV: The Woman Who Fell to Earth) However, she would discover that she was not, in fact, Gallifreyan at all, but rather the Timeless Child, an individual from another universe or dimension who'd ended up stranded in N-Space and was discovered and adopted by a Shobogan named Tecteun. From the Timeless Child the Shobogans gained the ability to regenerate, and eventually the Doctor had their memory erased and was regenerated into a baby or young child, becoming the young First Doctor. (TV: The Timeless Children)

Other universes
In one universe, when the Doctor was young, his mother and father went on a trek in the mountains with him. The Doctor's father and mother owned a summer house on the other side of Kasterborous. He remembered that her name was Penelope. (PROSE: The Infinity Doctors)

In the Obverse, the Doctor's mother was a mermaid. (PROSE: The Blue Angel)

Appearance
The First Doctor remembered his mother having red hair. (PROSE: A Big Hand for the Doctor) The Eighth Doctor also remembered that his mother had long red hair, and a cut-glass voice; (PROSE: The Gallifrey Chronicles) according to him, she resembled the third incarnation of Romana. (PROSE: The Ancestor Cell) At some point in life the Doctor's mother had grey hair. (COMIC: The Comfort of the Good)

Penelope Gate
Penelope Gate and Ulysses were intended by author Lance Parkin to be the Doctor's parents. Penelope's description matches the description of the Doctor's mother in The Gallifrey Chronicles, and, in the same story, Penelope and Ulysses are said to have a half-human son on Gallifrey. In The Infinity Doctors, the Doctor recalls that his mother's name is Penelope.

The Woman (The End of Time)
Russell T Davies has stated (in a March 2009 email reprinted in REF: The Writer's Tale - The Final Chapter, pages 622-623) that the unnamed woman, implied to be a Time Lord, in The End of Time was meant to be the Doctor's mother. However, he has also said that whoever the woman is can be up to judgment, and she could be someone else. Additionally, the depiction of the Doctor's mother in The Comfort of the Good bears a striking resemblance to the Woman in both physical appearance and Gallifreyan dress.

The Woman (Hell Bent)
Steven Moffat, the showrunner during the ninth series, has said that he has "no idea" who this woman is supposed to be, and leaves it to fans to decide whether they want her to be the Doctor's mother or the Woman from The End of Time is up to them.