To storm a castle, the invading army employed many different techniques. The most common way to make a castle fall was to besiege it. A castle could last many months if their supplies were enough; unluckily, there was not much knowledge in ways to preserve food.
The attackers would normally cut off any supplies to the castle (including rivers, commerce, farms). In addition to this, the invaders used catapults to throw dead bodies into the castle; spreading diseases. If a spy was captured outside the castle, he would normally be thrown alive as well.
Siege weapons were improved and widely used. Catapults were the most efficient siege weapons until trebuchets were introduced many years later. If a castle couldn't be besieged, the invaders used other methods to destroy or climb the castle's walls. The main method consisted on sending diggers who dug tunnels right beneath a wall (mostly corners since they were heavier). The tunnel would be strengthened with wood found from the surrounding area. After the tunnel was complete, the diggers would go to the surface. When they were all outside, fire was set to the wood; making the tunnel collapse. The walls were severely damaged when this method was employed.
To counterattack, the defenders would use water to see where the tunnels were being built. If a tunnel was found, the besieged army made another tunnel to intersect it. If such an intersection occurred, a fearsome struggle was bound to happen underneath the castle.
Archers were very ineffective if used from a lower height. Very big timber platforms were built in a matter of days to help the invaders gain some height for their archers; making them more effective. The crossbow was introduced in the XV century. It was so effective in killing people that the church prohibited its use for a brief period of time. Quarrels (crossbow's ammunition) could go from one side of a person to the other (even if he was wearing plate mail).
Making a castle surrender was a very difficult task. Sometimes months were not enough; which meant that an allied army could come to the defender's aid originating a double-front battle for the invaders.
Some of the most common ways to storm a castle were with the following devices:
The Battering Ram was mostly used against doors and thin walls. It consisted of knocking a chopped tree against its target with the strength of 6-10 men. Older Battering Rams had a roof made of iron to stop projectiles and boiling water.
Pickaxes were used against older castles in which the wall were very thin. They could make a hole through a wall in a matter of days. Even though effective against timber and thin stone, they were useless against bigger castles.
If a castle's door was made of wood, pickaxes could destroy it very hastily.
Diggers were used to destroy castle walls. Normally dug near a corner, the tunnel would be destroyed making the wall or tower collapse.
The most common and effective way to attack a castle was with siege. A castle would normally last many months. Food was the most common reason of the defenders' fall; many ways to preserve food were idealized. In some castles, the lord would get rid of all the women, old men and kids to preserve food. Usually they escaped through a tunnel, but sometimes no such tunnel existed and they had to stay inside the castle hoping for mercy.
The invaders surrounded the castle firing projectiles at the defenders.
Trebuchets and catapults were the most common ways to not only besiege a castle, but also spread diseases by firing dead bodies. Projectiles included: cows, huge rocks, dead humans and rotten meat.
Fire was very effective against timber castles. When stone castles started appearing, fire was practically useless. Attackers would shoot thousands of fire arrows in hope of burning anything inside (furniture, huts).
Also common, ladders were used against smaller castles. At a specific time, the attackers would send their army with many ladders hoping to climb the enemy's wall. This technique rarely worked since the defenders could throw objects--or boiling water--at their victims.
To conclude, attacking a castle was very difficult. Almost invariantly more attackers than defenders died because the castle itself was a formidable protection. Unfortunately for the defenders, food was scarce and they depended on a relieving force in order to survive.