The Great War started (and ended) on October 23, 2077 when nuclear weapons were launched by all the nuclear-capable nations of the world (mainly the United States, China and the USSR ). The war lasted for two hours, until the last atomic bomb felt. Then, the world fell into darkness.
Fortunately, some of United States inhabitants were able to survive the nuclear holocaust by sheltering into big Fallouts vaults, like, for example, Vault 13 and Vault 15. Those Vaults opened years later, letting the refugees spreading into the wasteland, a radiated and sterile earth.