Catapults are effective siege weapons which can be recruited from either the Siege Camp or the Engineer's Guild. Note however that regardless of which building you choose to recruit this weapon from, you must have an Engineer's Guild placed in your Castle to be able to recruit one.
While Catapults are primarily aimed at attacking walls, towers, gatehouses and other parts of a player's castle, they can also attack and destroy wooden buildings, siege camps and even units. You can also set your Catapults to fire diseased cows into an enemy Castle as well.
To do this, select the Catapult and then left click on the Cow icon. Note that this is only possible when you have at least one Dairy Farm placed in your Castle, with at least one cow. Changes will take a few minutes to filter through.
Catapults are relatively effective and accurate in dealing lots of damage against your intended targets, although like all weapons and ranged units, the longer the distance the less accurate they generally are. They will also struggle under lots of missile fire by ranged units, so make sure you use some Cats to protect your Catapults.
You can recruit Catapults from the rank of Baron. You will need 200 gold as well as 2 peasants from the campfire before you can recruit a Catapult.