A shotgun, also called a "scattergun", is a firearm designed to shoot a multitude of lead or steel balls of varying sizes. This makes it very effective at hunting game, greatly increasing the weapon's hit potential. However, the accuracy and power found at close range diminish rapidly as you get farther away from the target. Shotguns designed for military use are often called "combat shotguns".
This weapon deals the same damage in a non-critical hit as The Terrible Shotgun, but has a much smaller ammunition capacity (2 round chamber). Critical hit damage outside of V.A.T.S., however, is much larger due the larger number of pellets.
The name is most likely a reference to the commonly known idiom "shooting fish in a barrel" or a method of fishing known as "shotgun fishing" - in which a fisherman simply fires a large rifle or shotgun into a pond, using the shock-waves to kill fish.