Rebellions are when a city gets too low on happiness, resulting of a rebel faction popping out of the original. These rebel factions will have a leader and an army to capture whatever city it is revolting from. If you defeat them, they either flee or get wiped out. The next log will be about sieges.
This is a screen from turn 218, where a rebellion have broken out in a Floralian province. A army is defending the city the rebels are trying to take, lead by one Lucius. The garrison has the faction heir inside.