The Elders were three old men in charge of Gavaldon for the first four books of the series, but were replaced by a mayoral system after they were revealed to be corrupt villains.
The School for Good and Evil
The Elders wouldn't let Stefan remarry Honora without Sophie's blessing, but after Sophie was kidnapped they decided to marry without it. They also distrusted Callis and thought that she was a witch.
A World Without Princes
After Sophie and Agatha's wishes at the wedding, attacks from the Woods started destroying Gavaldon. In response, the Elders planned to give Sophie to the attackers, but told Agatha and Sophie's family that they were planning to move her to safety.
The Last Ever After
After Sophie and Agatha returned to the Woods, the Elders condemned them as witches, and would hang them if they returned to Gavaldon. When the Elders find and capture Tedros, Agatha and Callis, they find out that Tedros was behind the attacks that destroyed Gavaldon, and try to kill them. However, they only kill Callis, who uses her magic to help Agatha and Tedros escape, sacrificing herself.
Quests for Glory
The Elders' power is taken, in favor of a fairer system where a mayor is elected by the people.