Resignation was the act of quitting one's office or position.
The President of Gallifrey was able to resign and appoint their successor during a ceremony known as Presidential Resignation Day. (TV: The Deadly Assassin, AUDIO: Desperate Measures, etc.)
In an alternate timeline, Roger ap Gwilliam almost immediately resigned his position as Prime Minister of the United Kingdom after being elected, thanks to Ruby Sunday positioning herself such that he could hear the words of her counterpart. (TV: 73 Yards)