The Time of Awakening was a period in Vulcan history when the violent and warlike Vulcan race switched to more peaceful ways. It occurred sometime during the 4th century of the Earth calendar, although some Vulcans had already begun to mentally train themselves to suppress their emotions in the middle of the 3rd century. (VOY: "Random Thoughts")
The change was brought on by a Vulcan named Surak, who preached a philosophy of using logic to control emotions, as well as total pacifism. He was effective in spreading his teaching across Vulcan, though he encountered heavy resistance from many Vulcans, especially "those who marched beneath the raptor's wings", even starting a nuclear war. Although Surak died of radiation sickness at Mount Seleya shortly before the last battle against these factions, his writings were preserved in the Kir'Shara.
In the end, a sizable group of Vulcans eventually left the planet, becoming the ancestors of the Debrune and the Romulans.
It should be noted that there is no actual proof that these Vulcans were actually rebelling against Surak; all that is canonically known is that a group of Vulcans left the homeworld around this time, but whether or not they opposed Surak's teachings has never been officially stated on-screen. It is however likely that the faction Surak referred to as "those who march beneath the Raptor's wings" were the precursors of the Romulans, given that the Romulan Empire's emblem is a raptor, and their violent, warlike culture.
A deleted scene from "Chosen Realm" would have established that even after the embrace of logic, Vulcans continued to war over which exact interpretation from the teachings of Surak was correct. This resulted in a cycle of violence, with faction trying to suppress others, only to find that this was impossible and the other faction eventually retaliating. The cycle was eventually broken by the embrace of non-violence.
In TNG: "Gambit, Part II", it was referred to as the "Time of the Awakening," with a definite article.