a siege is when knights go to force another castle to surrender by having them use up their resources. Sometimes sieges can take months or even years. The knights will not go back until the castle surrenders. Then they set up a campsite surrounding the castle walls. The tents are loaded with strong weapons, food, water, and armor. The reason why they did this is because when they run out of food and supplies, the enemy will want to leave the castle for more resources. The knights can make them surrender because it's much easier to do this rather than attacking right off the bat.
A Mangonel is similar to a catapult. It gets loaded with a projectile and the projectile gets thrown by the arm of the mangonel. Usually, the Mangonels project a large rock or a payload. The mangonel will destroy things over the castle walls causing a lot of destruction to the castle.