What is the gun sling called where the gun drapes across the chest?

Answer 1

It is called a bandellero.

what the previous answerer was thinking of was a bandolier, which holds ammunition. A similar strap that holds a larger gun like a rifle or shotgun is simply called a sling. What you're looking for is probably a CQD sling or a Tactical sling. If its a holster where a pistol rests on the hip but the strap goes over the shoulder, its sometimes called a baldric. If the strap goes across the chest and the pistol is under the arm its called a shoulder holster. I think that one of those should cover it.