For the Vulcan event, please see Time of Awakening.
The Great Awakening was a series of events on the planet Argelius II which took place in the 2060s. During this time period, Argelian society instituted a series of reforms which eliminated violence from their society, adopting a pacifistic nature.
The Argelian empathic contact predated these occurrences. In 2267, Prefect Jaris mentioned this fact when suggesting that they use that method to determine whether Montgomery Scott had murdered Kara. (TOS: "Wolf in the Fold")