Imperial-Renegade Dalek Civil War

The Imperial-Renegade Dalek Civil War originated less than a day after the forces of the Supreme Dalek woke Davros from suspended animation.

Davros possessed a small injection device which could instantly convert both Humans and Daleks to his cause. He used this to build a small army of Daleks and Human replicant troopers, which he then pitted against the Supreme's forces. All of Davros' forces were destroyed, either by loyalist Daleks or the Movellan virus. (DW: Resurrection of the Daleks)

Davros escaped to the planet Necros where he began creating a new army of Daleks from human tissue. The Daleks led by the Supreme Dalek considered him a criminal and retrieved him from Necros. He was transported to the Dalek homeworld of Skaro to stand trial. (DW: Revelation of the Daleks)

While on Skaro, the Doctor rescued Davros and transported him in his TARDIS to Spiridon. The Doctor had a plan in mind to set Davros' faction, later known as the Imperial Daleks, against the Supreme Dalek's followers, later known as the Renegade Daleks. (DWM: Emperor of the Daleks)

The Renegades almost certainly did not use that term to refer to themselves.

The war ended after a final battle on Earth in 1963. Davros's Imperial Daleks defeated the Renegades with the aid of the Special Weapons Dalek. Subsequently, and further pursuing the plan he had already developed, the Doctor destroyed the Imperials using the Hand of Omega. The Doctor convinced the last of the Renegades to destroy itself. (DW: Remembrance of the Daleks)

For more details, see Shoreditch Incident.