What were the expectations of a medieval castle or fortress hoping to break or outlast a siege?

The basic idea was to make the expense of besieging more costly than it was worth. It is very difficult to take a castle. There were castles with a crew of less than 20 armed soldiers in it that successfully withstood a siege.

The same idea works today too: an armed man with a machine gun (32 rounds) can keep a large crowd under control, because very few people want to be amongst the 32 possible victims.

Your question is very broad. Castles vary from simple wooden motte and baily structures to very large professionally designed stone castles, such as Krak Des Chevaliers. William of Normandy had a large number of prefabricated motte-and-bailey castles with him on his invasion fleet. They were erected within days after the conquest.

Early castles were mostly like that: a ditch quickly dug around a large house, with the earth as a rampart with wooden stakes in it. If you think 'hey, that looks somewhat like a Vietnamese firebase, you are not mistaken. From these strong points the country was controlled and ruled.

How to defend a castle is also a broad subject. Most assaults in the early middle ages were small local affairs, no more than cattle raids by the neighbors. Others were large semi-professional armies that had siege equipment (England conquering Wales and Scotland, for example).

Now, to make it less complicated: take a prefab castle from William the Conqueror as an example. It was quickly erected by unwilling locals under the control of the Normans. They would lack almost everything: very little food to withstand a siege, almost no resupply of weapons. All they had was a safe place to stay for the moment.

Here the odds were in favor for the besieger, but ... it was (usually) enough. Why? Because the besieger had to gather a lot of men and materials to lay the siege. While he was doing that, the besieged weren't idly waiting around. They would strike out to prevent just that.

The besiegers would risk begin attacked from several different directions. Those small prefab castles were build all over already conquered parts of England. They offered a kind of mutual protection, like bunkers did during the world wars. (Without artillery, of course.)

To to answer your questions:

1- What would give you hope?

• The knowledge that your overlord probably wasn't waiting to see his property taken away. If you can hold out for a couple of weeks, the enemy will be attacked in the rear.
• Disease and hunger are now your allies. You have (likely) something to eat, a decent place to sleep and hopefully some toilet facilities. The besiegers probably not. They have to camp out, dig holes to crap in, often close to where they drew water (this really happened). They very likely run out of food in a couple of days and/or get dysentery. Later during the siege that changes, but for now it works in your favor.

2- What terms would you get?

Depends on how much effort the besiegers had to put in. Usually the terms could be fairly lenient if you surrendered immediately. If the besiegers had to take your castle by storm, don't expect any mercy. You're a dead duck. This was the common rule of warfare of the day.

The enemy didn't like to put in a lot of effort, and let you walk out afterwards: "was a nice game boys; see you next time!". That's what you do in a board game. The more effort he had to put in, the less lenient his terms would be.

• If you surrendered on first request you might be allowed to leave alive with some possessions, and civilians.
• If the castle had to be stormed, especially when that storming caused a lot of casualties amongst the besiegers: every surviving man, woman and child would be raped, tortured and killed.

I don't believe there is ever the intent to hide in a castle until a siege is broken by the people within the castle. The castle is meant to hold an area in the face of an overwhelming enemy until a force to lift the siege can be brought about. There may be a few examples where a port city (if not blockaded) could maintain a castle garrison for an indefinite amount of time, but it's not frequent. Moreover, the Romans proved time and time again that a tenacious siege with superior equipment could drop any fortification long before that cities supplies ran out. Should also be pointed out that several citadels and fortifications were built to siege cities, not protect them.

Garrisons are not cheap. It costs money to maintain a standing army for defence, and in many cases this expense was far too much to bear. The Romans solution for this was the Limitanei. Originally they referred to professional soldiers that resided in frontier towns prone to invasion, often Roman men that would marry locally and create their lives in the Frontier town but sometimes locals recruited to the Roman military. By around the 6th century, the Limitanei were part time soldiers and militia at best, referred to as 'frontier armies'. They manned the fortifications in the event of an invasion. The second line of Roman defence were comitatenses or palatini, which you would consider the more traditional military force. These were the professional soldiers that used the speed the vast Roman road network allowed to respond to these invasions and lift the siege on frontier towns.

A castle is rarely meant to repel an invasion. A castle is meant to protect an area with a relatively small garrison (at a small cost) by allowing that Garrison to retreat into that castle and wait for friendly re-enforcements to lift the siege. If there was no army capable of lifting the siege (castle was alone), then it was a matter of costing the invading army more than it would gain by taking the city.

what might give you hope to win

You were waiting for the much more expensive and better trained army of your Lord to come relieve you before the attackers could overwhelm you (if they could at all). You were never expecting to defeat and drive off the attacker yourself.

Sourced: http://www.romanarmy.net/Latearmy.shtml - much of it is there, this in particular:

Defence in depth

In any case, they were essential to the new system of Imperial defence. Rome now recognised that preclusive security could not be achieved; instead the assumption was that the barbarians would be able to penetrate Roman territory, and it was on Roman territory that they would be dealt with.

The presence of the limitanei was intended to deter invasion in the first place, but if it did happen, their role was to slow down the enemy advance and to siphon off their troops by holding newly constructed strongpoints known as 'burgi' ( which also acted as resupply stations for the mobile field army), fortified towns and other heavily fortified military sites along lines of communication. This would help protect the local populace, deny food to the invader, and most importantly, slow down enemy penetration, giving the field army time to arrive and confront them in open battle.

The main condition was sufficient food and water supply, and of course sufficient number of people to man the walls. Castles were actually very effective tool of defense, and in many cases the attacking army would just bypass them without a siege. But if the attacking army had enough people, enough time and good sources of supplies, then the only hope of the besieged was a help from outside. Experience shows that any stronghold could be taken, with sufficient time and resources.

There were three basic ways that a city could hope to survive a siege. They were 1) being relieved by a friendly force 2) outlasting the besiegers, and 3) "self relief," which was pretty rare.

If a besieged town was part of a larger polity, the greatest expectation was for relief by friendly forces superior to the besiegers. This held true for the Protestants at Derry in 1689, but not for the Gauls at Alesia in Roman times, even though both the besieged and the relieving forces outnumbered the besieging Roman forces.

The conventional expectation in Medieval Europe was that if a city could hold out for one year, it could outlast a siege. This was cited by any number of medieval writers, but the one I know best is Machiavelli's, "the Prince." A well prepared city would have food and drink and working materials for its citizens for a year. They would also hope that the weather and other elements would wreck more havoc on the besiegers. This multi-month defense eventually worked in eastern Europe against the Mongols, who could not resupply their horses and their "composite" bows as well in Europe's forests as on the steppe. An exception to this one-year rule was Carthage, which held out for three years, but eventually succumbed to Roman attack. As at Alesia, the Romans managed to defy medieval wisdom.

Self-relief was a pretty rare event.That's mainly because an army that elected to suffer being besieged was typically outnumbered (Alesia was a notable exception.) So a surrounding army could "entrench" and take the defensive as muc as the besieged. And the besieged armies had many of the difficulties of other surrounded groups, such as being cut off from help or resupply, with the notable exception that the city or fortress prevented it from imploding. A notable case of self-relief was Crusaders capturing the city of Antioch, being besieged in turn by the Saracens, and breaking out a month later.

*The "one year" rule also applied to serfs seeking to escape their masters. The operative motto was "Statluft macht frei," that is "City air makes one free"applied to both individuals and cities.

Another way castles hoped to outlast a besieging force, certainly in medieval times, was disease and general hardship. The castle provided cover from the elements and a steady supply of cached food plus some basic sanitation (cesspits), while the besieging force lived off of the land, with minimal cover for foot soldiers. Consequently, the forces surrounding the castle were more prone to disease and the effects of inclement weather.

Once the besieging army had stripped nearby farms of all available food, they would have to devote more people to foraging further to stay resupplied. And in preindustrial times, humans were more prone to fatal diseases from exposure to the elements than they are today.

One of Napolean's methods for success was a sophisticated supply system to keep his armies fed and supplied while on the march, making them less dependent upon foraging for food.

There is no one answer to your question. The outcome depended not only on the strength of the fortress, but also the force ratio, the capability of each side to endure a war of attrition (i.e. logistics), the determination of each side to win despite the costs, and superiority of weaponry. See, generally, [Kress and Talmor, "A New Look at the 3:1 Rule of Combat through Markov, Stochastic and Lanchester Models," Journal of the Operational Research Society, 50 733-44 (1999)].1

Historically, armies (and their navies) preferred to avoid direct confrontation with a fortress (both before firearms and afterwards). Going over or around the fortress is preferential, with modern examples being the German avoidance of the Maginot Line, and the US invasion of Kuwait.

If that were not possible, then armies would prefer a siege of the castle or walled city (or an entire country). Josephus tells us that but for Israelite zealots destroying food stores in the walled city of Jerusalem -- to force Israel to fight -- Jerusalem might have been able to withstand a long siege and would have the motivation to hold out. The Romans, however, having to answer to Rome and explain why the Legion was unable to put down the revolt, had little motivation for a long siege.

Superiority of weaponry was a major factor. In the pre-gunpowder days, the relative strength of spears and shields were key factors as metallurgy evolved. In the 19th century, especially during the American Civil War, the Union Army's acquisition of guns that could be reloaded and fire more rapidly than the guns of its enemies, played a huge role in its victory.

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Castles were very hard to take and required several months of siege and a numerically superior force to take. It was really a logistical battle between the siege force and the castle. If the siege force ran out of food, which was common given the logistics of most armies at the time, it would have to withdraw and end the siege. If the attacker was smart they would only lay siege if they had adequate supplies and could fend off any counterattacks.

The Siege of Orléans in which Joan of Arc fought was a good example of a castle/fortress which successfully outlasted the attackers until help could arrive.