Civil war is devastating to any empire. It is triggered by the loss of your capital. Up to half of your cities will rebel and declare allegiance to a new (AI) leader, who will loot half of your treasury and retain all of your scientific advances.
Players remain in the civil war state for just one turn, after which the empire enters a state of anarchy.
The capture of your capital does not always lead to civil war. If you have treated your people with kindness, you are more likely to retain their loyalty. Each city that is celebrating reduces the chance of civil war, while each city in disorder increases the likelihood.
In addition, the form of government directly contributes to the chance of civil war. Governments with universal franchise are far less likely to revolt than those more despotic in nature.
The number of cities an empire needs before it can erupt into civil war is 10. That is, empires with fewer than 10 cities are immune from civil war.