Both terms, 9mm and 9mm Luger, represent the same ammunition, which is officially classified by the Sporting Arms and Ammunitions Manufacturers’ Institute (SAMMI) as the 9mm Luger. The 9mm refers to the diameter of the bullet and Luger comes from the Luger semi-automatic pistol, the firearm the ammunition was initially created for (designed by the German inventor, Georg Luger).
The ammo is also referred to as the 9mm Parabellum, which originates from the manufacturer’s motto: Si vis pacem, para bellum (“If you seek peace, prepare for war”).