Shots refers to the number of times a ranged unit may fire per battle. Every time the ranged unit shoots during combat it loses a shot. If the unit has a special ability to shoot twice or receives a morale, it loses two shots instead of one. Although the Ballista says it has shots, it cannot run out, and goes into negatives instead of becoming unable to shoot.
Ammo Cart provides unlimited number of shots for ranged units as long as it is in one piece. The shots start to decrease right after the Ammo Cart is destroyed.