Thirteenth Doctor

The Thirteenth Doctor was an incarnation of the Time Lord known as the Doctor and the second incarnation to come from the Doctor's second regeneration cycle.

She was notably the first female incarnation of the Doctor, as all of her previous incarnations had been male.

A day to come
When the Twelfth Doctor broke his toe, Clara Oswald suggested regenerating to heal the injury, but he berated the idea as a waste. (PROSE: The Blood Cell)

While suffering from the common cold, the Twelfth Doctor, overreacting to the illness, considered the possibility of regenerating. (COMIC: The Day at the Doctors)

After the Monk invasion, the Twelfth Doctor needed to know if his companion, Bill Potts, was under the control of the Monks, and deceived her into shooting him in a rage to see if she had succumbed to the mind control, secretly putting blanks in all the guns, and faking his regeneration to complete the illusion. He made it look like the process had started, but emerged as himself to show her that he had deceived her. (TV: The Lie of the Land)

Post-regeneration
After the Twelfth Doctor was gravely wounded by the Cybermen on the Mondasian colony ship, the regenerative process began. However, tired of "being someone else", the Doctor delayed the change for some weeks, (TV: The Doctor Falls) until an encounter with his first incarnation, Archibald Hamish Lethbridge-Stewart, and the Testimony caused the Doctor to concede that another regeneration would "[not] kill anyone". After taking a final look at the universe and providing advice to his next incarnation, the Doctor regenerated inside his TARDIS in an explosive fashion.

Staggering to the console in a daze, the new Doctor examined her face in a reflection, exclaiming "Oh brilliant". After pressing a button on the TARDIS' console, the TARDIS suddenly spiralled into chaos, caused in part by the Doctor's explosive regeneration, and the Doctor was thrown out through the TARDIS' doors in the confusion, with the TARDIS itself vanishing without the now-falling Doctor. (TV: Twice Upon a Time)

Appearance
This incarnation of the Doctor resembled a woman in her mid-thirties, with medium length blonde hair and hazel-coloured eyes.

Her hands were noticeably smaller than those of her predecessor, resulting in the ring she had worn in her last incarnation falling onto the floor upon regeneration. She was shorter in stature than her previous incarnation as well; the Twelfth Doctor's coat appeared at least one size too big for her. (TV: Twice Upon a Time)

First female Doctor
The Thirteenth Doctor is the first, in the show's history, to be played by a woman. Before Jodie Whittaker, though, the idea of a woman Doctor had been explored.

The idea that a female actor could take the role of the Doctor was first publicly introduced by John Nathan-Turner and Tom Baker in 1980. By Baker's suggestion, he told the press, "I certainly wish my successor luck, whoever he—or she—might be." Peter Davison was cast as the Fifth Doctor, but the idea remained alive.

The Ninth Doctor before regeneration, stated that he could become anyone, (TV: The Parting of the Ways) with the Seventh and Tenth Doctors both claiming that regeneration was a lottery. (TV: Time and the Rani, The Day of the Doctor)

The Curse of Fatal Death (1999)
Though a parody, The Curse of Fatal Death by Steven Moffat introduced another Thirteenth Doctor, played by Joanna Lumley. This Doctor, upon regenerating, immediately noted that she had "etheric beam locators". When Emma tells her that those are actual breasts, the Doctor says that she "always wanted to get [her] hands on one of these". At the end, she suddenly finds the Master attractive, and walks off with him, arms around each other's waists.

Exile (2003)
The Big Finish Doctor Who Unbound story Exile, though not set in the prime Doctor Who universe, starred a female Third Doctor, played by Arabella Weir. The story tried to establish that, in this universe, suicide was necessary for a "sex-change regeneration", which was also considered a crime by the Time Lords. The story also depicted this first woman Doctor as a failure, hiding from the Time Lords: an alcoholic with a boring job at Sainsbury's, and a dull life, void of adventure.

Later stories, particularly in the Steven Moffat era, would contradict the idea that changing gender in regeneration was anything out of the ordinary, and certainly lifted the veil of shame.

Build-up in Doctor Who
In 2010, The End of Time, Part Two had the Eleventh Doctor mistakenly think he had regenerated into a female form, immediately post-regeneration. He quickly realised that he was mistaken, on finding his Adam's apple.

Neil Gaiman's script for The Doctor's Wife (2011) was perhaps the first to make reference to a Time Lord living out their many lives as more than just one gender. The Doctor talks of the Corsair, a "fantastic bloke", who was also a "bad girl" in a couple of their lives.

In 2013, The Night of the Doctor had the Sisterhood of Karn offer the Eighth Doctor the choice of "man or woman", for his approaching regeneration.

Just the following season, Steven Moffat's Dark Water introduced audiences to, the first female incarnation of the Master. Series 9 next showed the regeneration of the General, in Hell Bent, from a male body to being a woman once more.

Series 10's World Enough and Time included a rooftop conversation, between the Twelfth Doctor and Bill, in which he's only "fairly sure" that his first incarnation was a man, as it was a long time ago. The Twelfth Doctor here claims that Time Lords are "beyond [the] petty human obsession with gender and its associated stereotypes". The stage was set. By the time Jodie Whittaker was announced as the actor to follow Peter Capaldi, it was well-established, to viewers and within the Doctor Who universe, that the Doctor could be played by a woman.

Other matters
This is the only known incarnation of the Doctor that is not known to have participated in the Last Great Time War.

She is also the first incarnation that is completely unknown to River Song; the last time she met the Doctor before her death, she encountered the Twelfth Doctor, the first in a new regeneration cycle, and had not been aware that he had lives past the Eleventh.