Sexism was the idea that all members of one gender were superior or inferior to another.
By humans
Sexism could be found throughout human history.
Women in the early 17th century were often suspected of witchcraft and burned or drowned for it. Otherwise, woman were frequently regarded as lesser than their male counterparts at best. Indeed, the Thirteenth Doctor remarked that this era was a hard time for women because "when we’re not being drowned, we're being patronised to death". (TV: The Witchfinders)
While visiting Regency London in 1814, Bill Potts was subjected to sexist (as well as racist) remarks from Lord Sutcliffe, who referred to her as a "creature" and demanded her to "show respect in the presence of [her] betters". The Twelfth Doctor responded by punching him in the face, knocking him down. (TV: Thin Ice)
Roz Forrester found it difficult to understand the racism and sexism that she experienced in Paris in 1919 since such attitudes did not exist between humans in the 30th century from where she came. (PROSE: Toy Soldiers)
One professor in 1963, Professor Garland, held that women were not suited to serious academic study, as "they tend[ed] towards emotionalism" and lacked the necessary "temperament". He promoted a far less skilled employee at the National Museum, Timothy, over Helen Sinclair. (AUDIO: The Red Lady)
Sexism could go both ways as Peri Brown believed that men were incapable of explaining simple mechanics (PROSE: Attack of the Cybermen) and River Song thought that it "looked weird" when a man was thinking. (TV: The Husbands of River Song)
The Doctor
The Twelfth Doctor was rather fond of the First Doctor, but was incredibly embarrassed by his original incarnation's frequent sexist remarks. However, the First Doctor was not truly sexist and only making the remarks to annoy his future self, as he did not like him to begin with. He stopped as he grew to like his future self more. (TV: Twice Upon a Time, PROSE: Twice Upon a Time)
The Fifth Doctor had a similar reaction when the First Doctor assumed that Tegan Jovanka got refreshments for his future self, the Fifth Doctor asking his companion to humour his younger self's quirks. (TV: The Five Doctors)