The Hybrid
The Hybrid was a Gallifreyan legend that said all Matrix prophecies predicted that a hybrid creature, thought to be crossbred from two warrior races, would stand over the ruins of Gallifrey and unravel the Web of Time, breaking a billion billion hearts to heal its own. Rassilon searched for information about the Hybrid. (TV: Hell Bent)
References
It was popularly supposed that the two warrior races would be the Time Lords and the Daleks. (TV: Hell Bent) Davros attempted to fulfill the prophecy by giving some of his Daleks regeneration energy taken from the Twelfth Doctor, transforming them into Time Lord hybrids. (TV: The Witch's Familiar) However, the Doctor later claimed the Daleks would never allow anything to be half-Dalek. (TV: Heaven Sent)
When Ashildr became immortal after being given a Mire repair kit, the Doctor mentioned that there was "a little piece of alien inside of her, so in a way ... she's a hybrid". (TV: The Girl Who Died)
The Twelfth Doctor claimed at various points to possess special knowledge about the Hybrid, going so far as to one point stating that the Hybrid was "me". (TV: Heaven Sent, Hell Bent) It was suggested that knowledge of the Hybrid prophecies from the Cloisters originally drove the Doctor from Gallifrey, and that the Doctor himself was a half-human half-Time-Lord hybrid. (TV: Hell Bent) However, many believed that the Doctor was bluffing. (PROSE: A Brief History of Time Lords)
In the Cloisters at the end of the universe, the Doctor declared that Ashildr, who at that point went by "Me", was the Hybrid, at which point she countered with the theory that either he singly was the half-human Hybrid or that, if the Hybrid could be interpreted as being more than one being, it could refer to the pair of the Doctor and his companion Clara Oswald. Just prior to losing his memories of Clara, the Doctor indicated that his reckless actions in trying to prevent her death made him the Hybrid. (TV: Hell Bent)
The author of the Time Lord Academy textbook on Gallifreyan history thought that the Doctor was bluffing about being the Hybrid, and the Hybrid was never going to be half-Dalek. He proposed other candidates: a splicing of Human and Mire; a girl part-Human and part-wolf; and two girls half-human and half-Time Lord, one made Lord through metacrisis, the other conceived in the Vortex. However, none of them quite fit into the prophecy. (PROSE: A Brief History of Time Lords)
In 2002, Sam Jones told Griffin about a Gallifreyan prophecy concerning a hybrid. She said the Doctor was a hybrid, his father a Time Lord and his mother a human, and that he was destined to unite the two races and "bring good old human niceness into their alien society". Though she thought she was bluffing to stall Griffin, the Eighth Doctor later asked her who had told her about the prophecy. (PROSE: Unnatural History)
Behind the scenes
Although Hell Bent, among other series 9 episodes, offered multiple possibilities as to the identity/nature of the Hybrid, showrunner Steven Moffat, in an interview for Doctor Who Magazine #504, confirmed that the Hybrid referred to the pairing of the Doctor and Clara, and retroactively linked this fact to the reason why the Doctor's TARDIS exhibited some hostility towards Clara in series 7. (This would also explain the hearts the Hybrid would break; the hearts would belong to the audience.)
