A simple stick of wood in the right hands…
From the beginning of recorded history people have been fighting and one of the first items they probably choose to fight with was a stick. One caveman picked up a stick from the ground and brained another caveman. Thus, war was born and the use of weapons against another person was born, as well. Many years later, the stick was changed and fashioned into a staff and the quarterstaff or short-staff was born.
Many cultures used a staff as a weapon. The English quarterstaff really became common during the mid-16th century, which is when we see the first uses of the word. The term quarterstaff refers to a staff of wood, 6-8 feet long. Sometimes these staffs have metal tips, called ferrules and sometimes they have points but certainly the most common is just a staff with uncapped wooden ends at both ends. The name quarterstaff might be tied to the way the staff was made (quartering a tree) or the way it was held (the hand being a quarter of the way in from each end).
Stick fighting is a version of martial arts that exists all around the world. Each culture has developed a version of stick fighting and certainly in Medieval and Renaissance Europe, stick fighting became very popular. There are books dedicated to teaching stick and staff fighting dating back to the 15th century in England. Staff fighting was something everyone could be involved in. The staff was easy to manufacture and could be very effect as a defensive weapon and deadly as an offense weapon. If you were looking to outfit the commoners with a weapon, the staff was by far the cheapest and the easiest to teach them to use.
Therefore, it is no jump in logic to find the quarterstaff being used in legends such as Robin Hood. Little John and Robin fight each other with quarterstaffs instead of Robin paying the toll to cross the river. Robin of course bests Little John but the important part is the fact that they fought each other with a traditional English martial art of stick fighting or staff fighting. A commoner and a noble both understood the techniques of this ancient martial art and though Robin won, Little John was certainly no slouch.