So, first off, the Middle Ages is a really long period of time and the technology involved in sieges changed extensively over the centuries. I know basically nothing about early medieval sieges so somebody else will have to contribute to that.

The simplest answer to your question is that medieval armies attempted to undermine or ignore the defensive walls of the city/castle they were besieging. Direct assaults were pretty rare unless the walls had already been compromised in some way or some form of technology had been brought along to get over them. There are a few core ways to achieve this state and make an assault viable, in a rough grouping they are: using sappers to destroy a section of wall, bombard the walls with siege engines, and/or take the walls with a siege tower.

Sappers are probably the area of medieval siege I know the least about, so someone else here might know more. Sappers were men who would approach the walls (usually under some kind of protection). They would then dig under the walls and set a fire there which would eventually heat up the rocks in the walls and cause them to become brittle and prone to shattering, which could bring down the wall section. Alternatively they could dig tunnels and dig out a tunnel under the wall which could then be collapsed, in theory bringing down the entire wall. Being a sapper was pretty insanely dangerous and it didn't always work but when it did it was a great way to cause a breach in a wall that could be assaulted by the main army.

Technology for bombarding a castle's walls changed dramatically over the course of the Middle Ages. I'm not particularly familiar with pre-Trebuchet methods though. The counter weight trebuchet was introduced during the High Middle Ages and was widely for several centuries. Trebuchets were built on site and building one was expensive and time consuming. They could eventually tear holes in walls but just as easily they could be used to make life inside the fortification so unpleasant as to force a surrender. Edward I somewhat notoriously once forced the garrison of Stirling Castle, who were attempting to surrender, to stay in their castle while he fired on them since his trebuchet had taken 3 months to build and he was damned if he wasn't going to get a few good shots off with it.

Towards the end of the middle ages gunpowder artillery began to replace trebuchets as the bombardment weapon of choice. While gunpowder weapons first appeared in the early 14th century they didn't become so effective as to be noteworthy until the mid-15th. Charles VII of France and the Valois Dukes of Burgundy are probably best known for their widespread use of gunpowder artillery.

Siege towers were giant towers on wheels that would be pushed up against a castle's walls. Soldiers could then climb the towers and onto the walls. These were pretty widely used in the Crusades (Jerusalem was taken in the First Crusade via siege tower) but they were also used in European warfare. The towers provided protection for the soldiers inside from enemy attack but they were also prone to being set on fire, which could prove disastrous for anyone inside of them.

There are a few other technologies I don't have any specific knowledge of how they were used (battering rams and ladders being the first to come to mind). There were also other examples of people taking castles without these technologies. Robert Bruce was remarkably good at going taking castles by subterfuge, usually by climbing up ladders into them at night while the garrison wasn't expecting it, but he is definitely an exception not the rule. In general you were better of negotiating a surrender rather than trying to take a castle by force.

The collected volume Medieval Warfare, edited by Maurice Keen, has a decent bit to say on siege warfare, both in general and in the sections on each era.

Pretty much any book on the Crusades will include a discussion of the methods used a the major sieges during that era (there were a lot), I generally recommend Thomas Asbridge's The Crusades.

I haven't read Jim Bradbury's book The Medieval Siege yet (it's on my shelf) but his book on medieval Archers is excellent so I expect that this one is worth a look as well.

Castles were built to be able to be defended by a small garrison against large numbers of enemies.

It was difficult to take them by assault.

An attacking power that wanted to try to take a castle might first try bribery ("Here are some really good terms and benefits if you surrender the castle"), or treachery ("If you open the back gate, or lower a rope ladder over the wall, you will be richly rewarded), or a surprise attack ("We will get some picked men in disguised as merchants, who will seize the gate tower and hold it until our concealed reinforcements arrive").

If none of these approaches were likely to work, and the attacker wanted to conquer the castle quickly, they would need to try some form of assault.

Assault could take several forms (one at a time or several together).

The attacker could try to scale the walls, either with ladders, or with a siege tower.

The attacker could try to break the walls or the gate, with fire or axes (if the walls were wooden), with a battering ram against the gate, with mining under the walls and then causing the walls to collapse, or with artillery (trebuchets, or later cannon) to try to smash a breach in the walls.

Sometimes, these assaults worked, but generally, (until the Italian wars in about 1500 when the French first employed a mobile siege train of heavy cannon) they failed.

Until the introduction of heavy cannon, castles provided a strong defense. Even if one line of walls was scaled or breached (often at great cost in lives to the attackers compared to the defenders), castles had further lines of defense.

The only really reliable way to conquer a defiant castle was to surround it, cut it off from re-supply, and wait until the defenders surrendered due to starvation.

Of course, this tied up large numbers of troops besieging the castle. Meanwhile, the castle's allies might be mustering a relief or at least raiding the supply lines of the besiegers and making life difficult for them.