This varies a LOT depending on where and when you are, the strength of the defender and the attacker, any troops left outside the walls, allies turning up and many other things.

Generally, in western warfare, the goal of the war was to compel your enemy to agree to your demands (which could include seeing him dead).

How the siege will play out will be determined by several factors.

1. Time.

Depending on how much time either side has to devote to the siege, it will play out very differently. If the forces in the defensive position have large supplies of food and fresh water, good quarters and not too crowded conditions, they have emple of time. They know their soldiers and horses will not be decimated by disease or starvation. This will affect how willing they are to sally (go outside the defensive position to attack their enemy), negotiate or surrender.

On the other side, the attacker is usually limited in time. He depends either on supply trains from his own territory (rare during this era, as infrastructure is usually not good enough to allow transport of large amounts of supplies very far) or from looting the local countryside. The longer his army remains in place, the more sucked out the local countryside will be. Peasants will flee, often bringing whatever food and livestock they can with them. Farms will be burned, grain storages depleted and so on. The attacking army will need to go farther and farther away to get supplies, which will make them more and more vurnurable to raids from any supporters of the defenders outside the defensive structure as well as any sally from the defenders. Other things that may restrict the time available for the attacker are allies of the defender coming to the aid of the defender, his army being needed elsewhere, untouched lands of the defender or his allies raising an army to attack the attackers, the potential hostile reaction of religious figures, the commoners, other men with power or his own army. The army may consist partially or exclusively of part-time soldiers or peasant militia that needs to get back to their farms or jobs by a specific time. Disease or starvation might hit the attackers harder than the defenders, making his relative strength to the defender go down, and many other things.

2. Strength

The relative strength of the attacking and the defending force will change how the siege happens a lot. Larger and more well-organised armies might have a dedicated siege train with trained and experienced engineers that could quickly build (or even brought with them) siege engines such as catapults, onagers, ballistas, trebuchets, siege tents, battering rams, siege towers, assault ladders or even cannons (which started appearing in sieges in Iberia and the Middle East during the late 1200s and were firmly established in Europe by the time the English used them in the Battle of Crecy 1346).

The relative strength of the two forces would change a lot how the siege would play out. If the defender consists only of local militia and a few men-at-arms, they mightbe content just trying to hold out until help arrives. If there's a small army in the defensive structure, they'll be trying to sally and raid the enemy camp, try to get cavalry through the enemy siege works to raid his supply lines and his foragers that are raiding the countryside for food.

An attacking force far stronger than the defending force might try a direct assault to end the siege quickly, while a force that is not as superior might take care to build extensive siege works of trenches, pallisades, siege engines and mines (digging under the defensive structures in an aim to collapse them) while an attacker not confident in attacking the enemy defences might just opt to wait until the defender runs out of food and surrenders.

The strength of the defensive fortifications will also play a big part in this. Moats prevent digging mines (as they will become filled with water and drown the diggers) and bringing siege engines up to the walls. High towers and well-constructed walls allow the defenders a vital range advantage with projectiles. Hidden gates and secret passages allows the defender to raid the enemy camp (to destroy siege engines, capture supplies, confuse and cause casualties) in sallies. Defensive rings allow the defenders to retreat to the next line of fortifications should the first one be ruined or fall.

3. Priority

How important is this place and what it contains, both to the defender and the attacker? If the attacker is after the King, and he's among the defenders, he's going to prioritise this defensive structure above most other things. If the main army of the enemy is inside the defensive works, that is going to give it high priority too. If it is a local knight, his half-a-dozen men-at-arms and two dozen barely armed peasant militia inside a watchtower, all of which are irrelevant compared to other, bigger fish, then you will see a much different siege.

How long the defender is willing to stick to defending the defensive structure will also affect the siege, which brings us to the next point.

4. Negotiations

During a siege in western Europe, there were often negotiations between the attacker and the defender. The defender could be offered very lenient terms if they surrendered - being able to march off with supplies, banners, weapons and personal wealth could be one. The trheat of putting everyone to the sword once the fortification did fall was common, and the defenders knew they risked this. However, the defenders would also need to show that they fulfilled their duty and did not surrender immediately, especially if there was something larger at stake. An attack and then resumed negotiations, when both sides knew better the strength of each other could happen.

So, now that we have established these basic rules, let us make a scenario with a few sieges, shall we?

King Edward Whatever of England decides that he shall rule Scotland. The time seems ripe - King Alexander MacWhoagain are facing a noble revolt and have a hard time putting it down.

King Edward musters his army - 4 000 longbowmen, 2 000 men-at-arms and 1 000 knights and mounted squires. Along comes his siege train of 500 engineers and carpenters. So he marches north.

The first thing he encounters is a small watchtower surrounded by a pallisade. The garrison, some 20 men under Michael MacRefusal will not surrender. However, Edward does not have time for this - Alexander is gathering an army to oppose him, neither is this small castle any high priority - even if left alone, it will be a minor nuisance at most, but Edward still stations 50 men-ar-arms to besiege the castle.

This siege will be very low-key. The English will simply wait out the Scots, and the Scots will hope that Alexander will be victorious so that the English will be forced to retreat south again.

The next place Edward encounters is the hamlet of Raiderborough, under control of Clan MacWestealstuff. It is a decent-size castle, a hamlet, a small port and a low city wall. The Clan MacWestealstuff are habitual raiders and they have a sizable force of light cavalry, some 500 that could harm Edward's ability to loot and resupply from the countryside, so they need to be locked in. The port would be a nice way to get supplies and reinforcements to the army quicker, so taking the hamlet is decent high priority. Edward stations about 2 000 men to handle the garrison of roughly 1 000 men - they get some 50 of the siege engineers and carpenters as well. The English have decent amounts of time, but the King would like to have that port soon.

This siege will be an active one. The Clan will attempt to raid the English camp to destroy siege equipment and get some of their light cavalry out into the countryside to disrupt the English ability to forage and loot supplies. The English will attempt to breach the walls to force the Scots to retreat to the castle, which surely is too small to house enough supplies to sustain such a large garrison for long. The English will attempt to mine under one of the major towers of the city wall to have it collapse while attempting to shoot a breach at another point in the wall with a few trebuchets that are assembled from the siege trains.

The third situation is Edinborough itself. It is a formidable castle, but not very strongly manned at the moment. King Alexander is in there, so it is top priority for Edward. Most of Alexander's army is up north fighting the noble revolters and are rushing south, however, it is being slowed by revolting noblemen that have negotiated with the English and are not allowing the army through.

This siege is top priority - Edward will win if he gets his hands of Alexander. It is also low on time - when the Scottish army does arrive, the English army does not want to be spread out in siege lines. The defender is relatively weak, but since Alexander is commanding himself, and he will lose his throne in any kind of surrender, negotiations are unlikely.

This siege will probably be a very active one, with the English rushing siege works for a huge assault to settle the war once and for all.