"War. War Never Changes."
But a little more info:
The Fallout Universe is set in our Universe but at some point around, during the 1950s things changed. Technology and Culture stayed in the 1950s. Culture stayed locked in the 1950s while technology advanced along the lines of how people in the 1950s visioned the future to be, hence 1950s era looking robots and vacuum tubes.
As time when on the World Started to run out of resources, such as petroleum and uranium. The United States controlled the last resources of petroleum. Around the 2050s the world started to fall apart. The United Nations fell apart, the European Commonwealth fell apart. Europe and the Middle East went to war with one another and then turned on themselves in what is known as the Resource Wars.
The Sovert Union still existed in the Fallout Universe but it was replaced as America's Number One enemy by China. China invades Alaska in 2066 to capture the Oil there, starting the Sino-American War.
America having already started a program to safe guard America from Doomsday continued the building of Vaults. The Vaults turn out to be an experiment. So much we don't know about them to this very day.
America invades Canada soon after China invades Alaska. America also invades Mexico and tries and fails to counter invade China.
America in by 2076 prefects Power Armour with the T-51b and breaks the stalemate with China. America soon starts to overrun mainland China. The Sino-American War is all but Over.
But On October 23, 2077 the Great War happens. World War 3. It is implied in Fallout 2 that China fired first. The world was destroyed and the World of Fallout was born.
I am not sure what else you would like to know? If you can narrow it down.
As I said there isn't much known about the Vaults, other then that they were experiments. There were 122 made and 17 control Vaults. The reason behind the experiments are unknown.