When you approach an enemy castle you have the option of besieging it. You will see how much food they have left and also what sort of equipment you need to besiege it. Ladders take only a few hours to build, but a siege tower may take several days. Increasing your Engineer skill will lower the construction time of both types of equipment. On the battle map itself there will also be some trebuchets, but these are merely decorative and cannot be fired.
As well as attacking directly you also have the option of waiting to starve them out. Depending on the size of your army your food supplies may well run down quicker than theirs do. If you spend too long away from the castle the defenders will dismantle your siege equipment, although their food supplies will still be depleted until they get a chance to replenish them.
If an NPC leader is waiting at the castle when you besiege it there is a chance that he will decide to leave, in which case you will have to fight him; unlike the castle battle itself, this will be a traditional plains conflict. If you attack the castle when his forces are still there they will be added to the defenders' already sizeable force, so you may like to wait until he moves on before attacking.
Attacking
Before actually leading the attack you will want to rearrange your forces (see Party management); generally speaking you should arrange your men in order of how effective they are as ground troops (mounted troops will not have their horses). When you attack the castle you will have to make your way up the ladder or siege tower while braving the enemy's shafts. At the top will be a group of four or more defenders (generally archers). There will also be one or more archers in each of the side tower areas. The remainder will be in the castle compound. It may take your troops some time to work their way up the ramp and down the stairs, especially if they are newly spawned. New enemies will always spawn just in front of the keep's door, so you will probably want to set that as a gathering point (with ). There will usually be a few archers on the far wall (the one you didn't climb up on) so you will want to head up there after the initial group in the courtyard has been defeated. Unlike normal battles (where the "round" will eventually end depending on the enemy's forces and your battle size setting) enemies will keep spawning in front of the doorway until all of the defenders' troops are defeated.
When the defenders are all defeated you will take the castle (any enemy heroes present will escape), along with the usual payment of gold and experience, and you will be given the usual exploration options. After you take the castle it will become the property of your faction (if you have one) along with its nearby village (if there is one). After a few hours or days the king will assign the castle and its village to his vassals, possibly giving one to you as your fiefdom (see Fiefs).