An army may lay siege to an enemy settlement over many turns. The defenders of a besieged settlement must sally forth or seek outside help to avoid starvation and defeat. Small settlements without fortifications may be directly attacked or encircled in an attempt to starve the enemy out. Larger settlements with fortifications generally require siege weapons to be built prior to attack.
The defenders of a settlement under siege will eventually starve and surrender unless they sally or are relieved by another armies. The number of turns a besieged garrison will hold out for is shown on its tooltip.
Province capitals are fortified. These need to be reduced or circumvented for the besieger to storm the city in battle. Siege weapons are usually needed by the attacker to reduce any fortifications.
A garrison without an encamped defending army can sally by selecting the settlement and then right-clicking on the sieging army.