A standard barrel, as pertains to firearms, is a barrel without modifications.
Along with numerous barrel modifications, Battlefield 4 allows use of the Standard Barrel on such weapons. As with the Underslung Rail, this implies that there are no attachments, and is expected to be the default option on weapons.
On most weapons, this would mean unsuppressed fire with full muzzle flash. Recoil will push straight back into the barrel; with most stocks set below the barrel axis, the point of aim will generally rise.
On shotguns, this also means that no chokes are used, and that the pellet pattern will be wide and circular.