Military historians, when talking about the ancient and medieval periods of the Western world, typically use the term light infantry to refer to any sort of lightly armored or unarmored foot soldier who makes use of a projectile weapon.
The exact equipment of light infantrymen varied across times and cultures, but the most common weapon, being essentially the simplest, was the javelin. Because the javelin required only one hand to use, it was sometimes coupled with a light shield, as in the Persian light infantry force that the Greeks defeated at the battle of Marathon in 450 B.C.E. More typically, though, light infantrymen went into battle with little or no protection other than the speed of their own two feet and maybe a dagger, to be used if the soldier absolutely couldn't avoid close combat. With the onset of the medieval period came the increasing dominance of the bow on European battlefields, although it had certainly been known and used in ancient times as well, especially by the seafaring peoples of the Mediterranean basin. One innovation that medieval warfare contributed in the area of light infantry was the invention of the crossbow. Although more expensive in material terms than a normal bow, the crossbow took less skill and practice to use and therefore somewhat recouped its initial outlay, if not exactly in kind. Other typical light infantry weapons were the sling, which was very popular on the island of Rhodes, and sometimes even a throwing axe or hammer.
The light infantry usually had a speed advantage over heavy infantry, mostly since they weren't carrying a hundred pounds of iron as protection. Thus, when able to use this advantage in combat by fleeing heavy infantrymen, the light infantry enjoyed supremacy over the heavy. Of course, there were ways to immobilize an enemy in order to do away with his speed advantage, but because of the versatility of the light infantryman in so many different kinds of terrain it was difficult to implement this strategy against him. The light infantryman was on more equal terms with the light cavalry than the heavy infantry, but light infantry still enjoyed battlefield supremacy. Because the light infantryman was so much more able to devote himself to the task at hand, killing people, due to the lack of a bucking horse between his legs, his skill with the bow was typically superior to the cavalryman's. Thus, although the light cavalryman has an edge in terms of speed, the light infantryman typically neutralized it by standing still to face his opponent, thereby bringing to bear his advantage, namely greater skill. Also, light infantrymen were the most valuable kind of soldier in defending a fortified position since they didn't have to leave the protection of the fortifications in order to hurt the enemy.
The only weapon system against which the light infantryman was particularly weak was the heavy cavalry. Because the heavy cavalryman was heavily armored against projectiles, the light infantryman had a lot of trouble hurting him. While the light infantryman was shooting arrows ineffectually at the cavalryman, the horse would allow the intervening space to be closed quickly, whereupon the heavy cavalryman typically made short work of the lightly armed and armored opponent. The light infantry's slight edge in speed on the heavy infantry avails them nothing in this situation, since the average human has a lot of trouble outrunning a galloping horse.
Economically speaking, the light infantryman was not too expensive to put on the field. Although he needed to be more skilled with his weapon as a matter of course than did the heavy infantryman, the nonexistent investment in the light infantryman's armor meant that the light infantryman was about as easy to field as the heavy.
Light infantry was seen most often in ancient conflicts as opposed to medieval, a fact that is probably most easily explained by the immense prevalence of the heavy cavalry during the latter period. After all, if the army you're facing is guaranteed to have a certain kind of soldier, why would you field men who would be guaranteed to be weak against it?
Of course, that's not to say that light infantry were either unuseful or unused during the medieval period. At the battle of Agincourt, King Henry V of England successfully defeated a predominantly heavy cavalry French force using mostly light infantrymen. He protected his flanks with forest and his front with sharp sticks that he had the archers plant in the ground in front of them in order to ward off the heavy cavalrymen's charge. He also made sure that heavy infantry was interspersed in the lines in order to provide additional support to his light infantrymen. This strategy proved enough to win the day against the odds, which just goes to show the importance of always carrying sharp sticks around with you.
All facts and most interpretations come from Archer Jones's excellent book, The Art of War in the Western World