Satan, usually known as the Devil, was, according to Toby Zed, "fury and rage and death". (TV: The Satan Pit) Oswald Danes once joked that, should the Devil walk the Earth, he'd certainly work in PR. (TV: Rendition) According to the Tenth Doctor, "more religions than there are planets in the sky" had devils, also known as beasts, including the Arkiphets, Quoldonity, Christianity, Pash Pash, New Judaism, San Klah and the Church of the Tin Vagabond. Neoclassic Congregational didn't have a devil per se, only "the things that men do", which the Doctor claimed was the "same thing in the end". (TV: The Satan Pit)
Many people were afraid of the Devil. In the early 17th century, the locals feared Satan so much they conducted many witch trials to drive him out of "possessed" humans. (TV: The Witchfinders) During an invasion of New York City in the 21st century, a Cerebravore used Hector's fear of the Devil against him. Hector believed that the Devil came for him because he knew his sins. (COMIC: Revolutions of Terror)
Origins of the myths
Several beings and groups were a possible source for the idea of Satan.
The Dæmons, a horned race from the planet Dæmos, were believed by the Third Doctor to be the inspiration for Satan based on their experiments on humanity. (TV: The Dæmons) Likewise, the Demoniacs were another species believed to have influenced human society. (PROSE: Mean Streets)
The Beast said that he was the origin of, if not the actual Satan, and all similar entities on all worlds. (TV: The Impossible Planet / The Satan Pit) The Tenth Doctor found the Beast, in particular his claim to have existed before the universe, a challenge to his own belief system. (TV: The Satan Pit) He did eventually refer to the Beast as "the devil". (TV: Planet of the Ood)
Scratchman, a being who fed off of entire universes, liked to present himself as the Devil, as he would often reach outside of his own universe to make Faustian bargains with the denizens of another universe. (PROSE: Scratchman)
On other occasions, entities were mistaken for Satan.
The Fourth Doctor claimed that the appearance of the Devil on Earth was inspired by an alien species conducting a survey mission on the planet. He reflected on this disdainfully, citing it as an example of cultural appropriation. (PROSE: Scratchman)
In 1928, when Angelo Colasanto learned of Jack Harkness' immortality, he thought he was the Devil. Jack was captured and killed repeatedly, though some thought he was a miracle instead of the Devil. (TV: Immortal Sins)
In his speech to the Clerics in which he told them that the Eleventh Doctor was a living breathing man, Colonel Manton said that the Doctor was not, among other things, the Devil. (TV: A Good Man Goes to War)