The Divergence were a race that would have risen to power after the Time Lords. Rassilon didn't want to commit genocide, so he locked them away in a time loop to be trapped until the end of time. "Divergence" was a name given to the creatures by Rassilon. It is unclear what their actual name was, if they had any at all.
Biology
Rassilon did not know whether the Divergence were a race, species, or an entire genus. (AUDIO: Zagreus)
History
In Cardington, 1951, Doctor Stone and Reverend Townsend opened a doorway to the Divergence. This failed and killed Stone and Townsend.
At the end of time, approximately sixty billion years in the future, they nearly did get out. Uncle Winky's Wonderland was built over Rassilon's lab, but Uncle Winky didn't open the door when he should have. (AUDIO: Zagreus)
The Divergence got their revenge on Rassilon for destroying their future when he was banished into the Divergent Universe and they subjected him to the torments of the cycle which he had created eighty-four times.
Eventually the cycle somehow caused the Divergence to wither and die but they were then absorbed by the creature known as Daqar Keep who subsequently embodied all their power and assumed their authority in the Divergent Universe. (AUDIO: The Next Life)