Racism was the idea that all members of a race were superior or inferior to another race. An individual with this belief was known as a racist. Some words used to humiliate other races were also described as racist. (TV: Remembrance of the Daleks, Human Nature/The Family of Blood)
By humans
While visiting Regency London in 1814, Bill Potts was subjected to racist (as well as sexist) remarks from Lord Sutcliffe, referring to her as a "creature" and demanding she "show respect in the presence of [her] betters". The Twelfth Doctor responded by punching him in the face, knocking him down. (TV: Thin Ice)
While Martha Jones was working as a maid at the Farringham School for Boys in 1913, Hutchinson made several racist remarks to her. (TV: Human Nature/The Family of Blood) She also feared that she would be carted off as a slave in Elizabethan England. (TV: The Shakespeare Code)
In 1941, during the Second World War, 2008 native Toshiko Sato was insulted by Americans in the Ritz dance hall because she was Japanese. (TV: Captain Jack Harkness)
In 1963, a sign hanging in the window of Mrs Smith's Shoreditch boarding-house read "NO COLOUREDS", which Ace seemed to be disturbed by. (TV: Remembrance of the Daleks)
The house of Ace's friend Manisha Purkayastha was firebombed by racists in 1983. (TV: Ghost Light)
In 2009, when Wilfred Mott remarked he was saved by the cactuses, the Tenth Doctor replied with, "That's Cacti", to which the Vinvocci exclaimed that it was racism. (TV: The End of Time)
Freda's house was burnt down by racists in 2069 because of her alien mother. She was saved by a member of Torchwood, who sent her back in time to 2009 via the Rift. (AUDIO: Asylum)
Roz Forrester found it difficult to understand the racism and sexism she experienced in 1919 Paris, since such attitudes did not exist between humans in the 30th century from were she came. (PROSE: Toy Soldiers)
Dahn-Ren briefly branded Bill Potts a racist after the latter expressed surprise at his blue skin. (TV: Oxygen)
Amongst other species
The Daleks' belief in their innate superiority and need to maintain their racial purity led to inevitable conflict between the Imperial Daleks and the Renegade Daleks. As Ace succinctly put it, "They hate each others' chromosomes. War to the death." (TV: Remembrance of the Daleks)
Time Lords had become hated by the universe due to the Time War. Cass instantly hated the Doctor when he told her he was a Time Lord, refusing to be rescued by him and telling to go finish destroying the universe, despite his claims to have never fought in the war. (TV: The Night of the Doctor) Even Clara Oswald was disgusted with the Time Lords, calling them monsters who had to hide out at the end of time because the rest of the universe hated them. (TV: Hell Bent)
According to the Fifth Doctor, the Pakhars found the term "rodent people" extremely offensive, even racist. (AUDIO: Nekromanteia)
Jeff Shrubb was accused of stirring up racism, sexism and homophobia on Olleril. (PROSE: Tragedy Day)
Behind the scenes
The Black and White Minstrel Show was known for its racism in the 1970s. Controversially, an Alpha Centauri made a guest appearance in 1972.