veteran and former paratrooper here. this is the badge for completing airborne school and becoming airborne qualified. edit: the pin is colloquially called "airborne wings" or "jump wings" which is what veterans will probably call it if it comes up.

That would be the Basic Parachutist Badge, awarded to any Marine, soldier or sailor that completes the three-week Basic Airborne Course of the US Army Infantry School at Fort Benning, Georgia.

Bronze stars attached to the badge denoting actual jumps into combat were unofficially worn since WWII; their official use was not sanctioned until relatively recently (December 14, 1983)

Sources:

US Army Quartermaster Foundation- Airborne Parachutist Badge