Private comes from the Latin word privus or perhaps privo that meant an individual person and later an individual without (deprived of) an office. That certainly describes a Private in our Army or Marine Corps. The term as a military rank seems to come from the Sixteenth Century when individuals had the privilege of enlisting or making private contracts to serve as private soldiers in military units. Before the Sixteenth Century many armies were simply feudal levies in which the feudal lords forced their serfs or subjects to serve.
Airman is a recent word that means somebody involved with flying. The Air Force gave that title to the members of its four lowest enlisted ranks in 1952.