The Master is the arch-nemesis of the Doctor in the world's longest-running sci-fi show.
In Canon[]
The Master appeared in early seasons of Doctor Who, each time with a devious and explosive plot to take over the entire galaxy, enslave its people, cause further explosions, and begin the cinematic trend of depicting evil via moustaches. Roger Delgado is the Master principally credited for this hitherto-unknown use of facial hair as a means of divining moral alignment.
In the course of his many run-ins with the Doctor in Classic Who, the Master carelessly managed to use up all 13 of his regenerations. Not content to let sleeping maniacs lie, the Time Lords resurrected him for unspecified reasons pertaining to the Time War. By the year one hundred trillion, he had ensconced himself in the form of Derek Jacobi and also on the planet Malcassairo, where he occupies himself building engines out of fairy liquid bottles and sticky-backed plastic, aided by his obligatory beautiful assistant.
Plot ensues, and the Master takes on the face of John Simm, who appears to be having far too much fun with the role, and thus not only makes a cracking evil scheming megalomaniac who actually succeeds in taking over the Earth and enslaving its population. In his free time, he likes to chain up immortals in his big shiny flying ship and perform unknowable and unspeakable deeds upon them. He plays the drums.
The Master later makes a return as the Mistress (played by Michelle Gomez), which made him—well, her—the first on-screen example of the Time Lord ability to change sexes upon regeneration.
In Badfic[]
In badfic, writers tend to greatly downplay exactly how much the Doctor and the Master hate each other and exactly how many times one has attempted to kill the other, all for the sake of bad slash. This has become much more pronounced since John Simm took the role, with many depictions presenting him as a quirky, fun-loving guy who just happens to be all slashy with the Tenth Doctor. The fact that that he's also a raving psychopath in this incarnation is similarly downplayed—although not always, which tends to lead to bad BDSM fic.
The Master and the PPC[]
It is not known if he is the star of Master Who in the mirror multiverse or if the Doctor is called "the Master" there.