The Doctor's mother

According to several accounts, the Doctor remembered having a mother, although her identity varied between accounts.

The Eighth Doctor remembered that his mother was a human (TV: Doctor Who, et al.) named Penelope Gate. (PROSE: The Infinity Doctors, The Gallifrey Chronicles) At other times, the Doctor remembered that she was a Time Lord (COMIC: The Comfort of the Good) or that rather than having parents on Gallifrey at all, the Doctor had been born from the loom of the House of Lungbarrow (PROSE: Cold Fusion, Lungbarrow) or arrived as the Timeless Child. (TV: The Timeless Children)

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) even after learning Gallifrey had not been lost in the Last Great Time War. (TV: The Day of the Doctor)

Whoever had been their mother in the childhood the Doctor remembered, accounts which purported that the Doctor had a secret history as the Timeless Child before having their memory wiped and beginning again as a child, showed that, prior to the mindwipe, the Doctor had considered Tecteun their mother. (TV: Survivors of the Flux)

Penelope Gate
Penelope Gate was an inventor from Victorian era England. She invented a time machine and traveled to 1996, where she was joined by Joel Mintz; later, the Seventh Doctor and Chris Cwej met her in 16th century Japan. (PROSE: The Room With No Doors)

Later, she met the Time Lord Ulysses, and they were married. They had a child, the Doctor, although they kept his hybrid nature secret from the High Council. (PROSE: The Gallifrey Chronicles)

When the Doctor was young, his parents went on a trek in the mountains with him. They owned a summer house on the other side of Kasterborous. (PROSE: The Infinity Doctors)

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 his father hold him up as a child to see the stars. (PROSE: The Eight Doctors)

The Doctor often remembered her long, red hair across his incarnations, (PROSE: A Big Hand for the Doctor, The Infinity Doctors, The Gallifrey Chronicles) and the Eighth Doctor often alluded to being half-human "on his mother's side". (TV: Doctor Who, PROSE: Alien Bodies, Grimm Reality, Unnatural History, The Shadows of Avalon)

Time Lord
By another account, the Doctor's mother was a Time Lord who wore Time Lord robes. (COMIC: The Comfort of the Good) Indeed, Peinforte believed that the idea of the Doctor's mother being human had been "disregarded" after reading through TARDIS Wiki. (PROSE: Lady Peinforte)

The Tenth Doctor saw a similar-looking woman kneeling next to Rassilon; she made eye contact with the Doctor before he sent Gallifrey back into the Time War. (TV: The End of Time) Later on post-War Gallifrey, an elderly woman entered a barn connected to the Doctor and saw the Twelfth Doctor, whom she quickly recognized as the Doctor and showed concern for. (TV: Hell Bent)

Tecteun
By one account, Tecteun was the Doctor's adoptive mother, taking up this role after discovering them as a lost child, waiting alone by a wormhole to a second universe. (TV: The Timeless Children, Survivors of the Flux) The Doctor called her Mother, though Tecteun regarded them as an experiment, (TV: Survivors of the Flux) having used the Doctor as a template for regeneration. (TV: The Timeless Children) She held little regard for the Doctor once they escaped her grasp, and felt the Doctor's travels came in the way of the Division's designs. (TV: Survivors of the Flux)

On discovering all this, the Thirteenth Doctor was angry with Tecteun for having "stolen" her, since neither could be certain what the Doctor had been doing where they were found. The Doctor felt she had been ripped from her history, deprived of the life she might have lived. (TV: Survivors of the Flux)

Other universes
In the Obverse, the Doctor's mother was a mermaid (PROSE: The Blue Angel) named Magda. (PROSE: The Dreadful Flap)

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)

The Woman (The End of Time)
Russell T Davies intended for the unnamed woman in The End of Time to be the Doctor's mother. However, he left it up to fan interpretation.

The Woman (Hell Bent)
Steven Moffat, the showrunner during the ninth series, has said that he has "no idea" who the woman in Hell Bent 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.

Other notes

 * Marc Platt stated that in Lungbarrow, he intended Leela and Andred's first child to be the Doctor (or, rather, the Other, who would one day be reincarnated as the Doctor), making Leela, in one sense, the Doctor's mother. However, while this was hinted at in the novel, it was never explicitly confirmed.