First, a timeline :

2000 Old Reckoning : The first world government was created after the World War III.
2015 Old Reckoning : Decontamination of North America is continuing, Europe is rebuilding.
2097 Day 1 of The Computer : The End.
Year 194 of The Computer : Year 194 of Alpha Complex. Year 194 of Universal Happiness. Year 194 of the New Order. Year 194 of Paranoia.

On May 4, 2097, a huge planetoid was first discovered near Saturn, and heading toward the Earth. Global state of emergency is declared. People are fleeing to the Earth's colonies. In USA, shelters are built in Des Moines and elsewhere in great haste.
On August 3, in Siberia, an old ICBM site identifies the planetoid as an incoming missile. A counter-strike is launched. A single missile without a warhead (removed 127 years before) exits the silo and heads toward San-Francisco.
San-Francisco - a jewel of the Age of Peace - was operated by a giant computer network known as "Alpha Complex".
"A Traffic Control sub-center of Alpha Complex tracked the missile and properly signaled for someone to take a look. No one came. (The population of San-Francisco was understandably preoccupied awaiting the end.)"
(stolen from a secret file titled "Alpha Complex - The Birth")
The sub-center alerted the Complex but it couldn't identify the missile, so it contacted the North American center, known as 'Alpha Prime'.

Alpha Prime couldn't find a mail missile with the incoming object's configuration, maily because his memory was moved to Des Moines. The computer center finally found something called "Civil Defense", a top-secret dating back to 1957. The planetoid was identified as a Soviet nuclear missile. Yep, that means WAR. Alpha Prime then contacted all the other complexes about an impending communist attack.
At this time, the planetoid entered Earth's atmosphere, striking in many places. Earthquakes shattered the Earth's crust.

The Alpha Complex Computer survived the disaster. It moved quickly to determine the extent of the damage and it found that it had lost contact with Alpha Prime and most of the other computers. To the Computer, there was only one plausible cause for all this mess : the city has been invaded by commies. The Computer started a resistance. This is the world of Paranoia

See also : (I should add them soon)

Alpha Complex's secret societies
Alpha Complex's mutant problem
Alpha Complex's organization