"Scattergun" is slang or informal terminology for a shotgun class of weapon. Briefly, a shotgun is a shoulder-fired weapon with an unrifled (smoothbore) barrel and is intended to fire one large or multiple smaller pellets of metal. If a single pellet (sphere or sabot) is used, it is of a large caliber and suitable for medium to large game at short to medium ranges. If multiple metal pellets or spheres are used in a loaded shell, it is suitable at shorter ranges for small game like fowl or rabbits. A shotgun variation not found in the game is "flechette launcher" which uses an explosive charge to throw a pattern of small aerodynamic darts in a cone from the smoothbore barrel. For soft targets (unarmored), this is very damaging to flesh, but useless for meat hunting.
In Star Wars: The Old Republic, the shotgun is a secondary weapon which has variable characteristics like endurance, accuracy, and so forth. Since it always has Cunning and Endurance at the minimum, it can only be equipped by certain character classes and companions. The shotgun can be purchased or found in green, blue and purple versions from drops as a pre-made with endurance+cunning plus other characteristics or found in orange customizable versions from vendors or crafters as a fully-moddable weapon with slots for barrel, mod, enhancement, and sometimes augmentation. In practice, it is sometimes treated as a "stick" upon which to add modifications to strengthen your character and given little or no use as a weapon.
Character classes using a shotgun/scattergun
The Scoundrel class uses the shotgun as a secondary weapon. Remembering that the Scoundrel has a short-range melee orientation, the shotgun is short-range only. Further, you can only use it from behind the target which often means your character must be in stealth or run-up from behind and fire quickly. To compensate, when used from stealth and beind the enemy, it can be devastating.
Companions using a shotgun/scattergun
Companions which use a shotgun as their secondary weapon include Guss Tuno, Risha, Doc, Zenith and Tharan Cedrax. Their use as a weapon does not appear to be restricted to being behind the target or in stealth mode, but is used in melee-range only. As in characters, players use it as a "stick" to hold mods to strength your companion. Companions can be equipped with pre-mades or custom shotguns (as described above).