The custom of saluting has always existed in all cultures. A salute is a gesture of respect to a person of superior rank.
Until the 18th century, junior officers saluted by doffing their hats. In fact civilians still salute in this manner. This custom probably goes back to the middle ages when a knight would raise his helmet visor or uncover his head before a lord.
After the 18th century, there came a change in the method of saluting, for a very practical reason! When soldiers fired their muskets, black powder used to settle on their hands, covering them with grime. And if they had to then take off their hats, it would spoil the hats! So by the end of the 18th century, the form was changed.
In some cases it involved bowing and in others kneeling or laying on the ground or gesturing with hand and arm in a special manner. An officer or soldier carrying a sword at the shoulder salutes by bringing the hilt to his mouth and then the point to the right and downward. This form of salute goes back to the middle ages.
The military salute of today — raising the right hand to the forehead or to the hat brim or visor is a recent innovation.