The Catapult is an Early Middle Ages artillery unit. It is the last unit to be unlocked in this age.
Compared to the Ballista, it gains +2 attack, giving it good attack values for an artillery unit. It doesn't gain any defense other than a slightly higher defense against heavy units, making it the weakest unit on a EMA battlefield.
It is most useful attacking Heavy Infantry. They are effectively used when put on hills and attacking Armored Infantry or other Heavy units, where they can cause serious damage over time.
The Catapult is followed by the Trebuchet of the High Middle Ages.
The catapult gains attack on hills, making it much more useful. They are especially vulnerable compared to an average artillery unit of another age. In the EMA mounted archers come into play, which are functionally Iron Age archers on horses. These mounted archers can generally reach Catapults on the first turn, effectively giving them a bonus against them. With one hit, Catapults will be reduced to almost zero health, and as a result will have very reduced attack capabilities. Even if they cannot reach Catapults on the first turn, they will still have to take more than 2 hits to be killed by them.
The best way to use Catapults is against Heavy Units, or against Archers. Since heavy units are slow, a fast unit can lead the heavy units on a chase while having the catapult hitting them every turn, but since Armored Infantry have very high defense, despite the Catapults bonuses, they will usually take 3 or 4 hits to kill. This theoretically can only allow a fast unit and one catapult to win a battle vs up to two full health heavy units.
They can be easily killed by Heavy Cavalry as well, due to their fast movement speed, and even if they are guarded by Light or Heavy Units, they can still rush past them and do massive damage against them.
Catapults are ineffective against the Mercenary, usually doing around one hitpoint if they are in cover, whilst the good movement speed of Mercenaries allow them to easily reach and kill them in two hits. Make sure to have heavy units holding off the light units before they reach the Catapults.
Overall, this makes the Catapults, a less useful Artillery Unit of this age, being only slightly effective against Heavy Units of its age, and weak against Light, Heavy and Ranged Units of the EMA.
Confirmed on PC: In battle sometimes the Catapult will quickly revert to facing a different direction than the way it was facing. For example, a catapult facing the enemy would flash to facing the other direction before reverting back to its original state. Does not have an effect on gameplay.