A siege of a castle is where people set up camp outside of the castle. If the castle runs out of food or water they fight or surrender. All of this can take months or even a year. The reason there were sieges was because people wanted the castle and to expand their power.
A catapult is a machine that can shoot far. It would shoot rocks and the rock could be lit on fire. The catapult is use to knock down walls and attack your enemies in battle. The catapults aim is really good but does not always hit the target. Sometimes the people that are protecting the castle use catapults in battle against the enemies.