When attacking player villages, your troops fight until one side is defeated. When the attacker is victorious, there is a change they will do siege damage or attempt to conquer the village.
Siege damage
Siege damage is done when the attacker wins the battle. To set up an attack to do siege damage, you need a Siege Workshop. This requires a Military District, so at least a tier 2 village (Town).
Siege Workshop
Depending on the level of the Siege Workshop, you can select 0 to 5 targets. A target is unlocked at level 1, 5, 10, 15 and 20.
Siege target selection with a Siege Workshop level 10 and no Siege Academy
For each available target, you also get a roll to damage the defender's Walls. This happens also if you don't select any targets. Each roll has a 76% chance to lower the Walls by 1 level.
Siege Academy
The Siege Academy can be built in a village with a Military District after researching Craftsmanship. The Siege Academy increases the damage to your siege attacks every 2 levels.
Without a Siege Academy, attacks damage buildings for 0 to 10 levels. A Siege Academy level 2 makes this 1 to 11 damage, up to 5 to 15 damage at level 10.
Loyalty and conquest
Each village has loyalty to their Kingdom. By attacking with a Noble, you can lower the loyalty of the village to convince it to join your Kingdom instead. When the loyalty of a village hits 0, the attacker conquers it.
Each surviving Noble will lower the loyalty of the village by 30. The more Nobles survive in an attack, the more the loyalty will be lowered.
Screenshot of an attack report where the village is conquered by the attacker
A village starts with 100 loyalty by default. A Town has 200 loyalty and a City 300. Loyalty increases with 1% of the maximum per hour, so 1 per hour for a village and 3 per hour for a city.