Light cavalry is one of the most tactically useful weapon systems of the ancient and medieval world. Their two prime attributes, speed and long-range attack, both complimented one another nicely such that in certain terrain, few other weapon systems could effectively oppose a light cavalry force.
The average light cavalryman would usually be mounted on an unarmored or lightly armored horse with himself armored similarly. For a weapon, most anything that had reasonably long range worked: javelins were the most common (being essentially the simplest), with bows, crossbows, and even throwing axes sometimes being used. Things would also be different when the time was considered. In the ancient period, crossbows were unknown and javelins were extremely prevalent in the west, while bows came to prominence above all other weapons in Europe with the onset of the medieval period.
The multiple advantages of light cavalry allowed it to enjoy a number of different uses in battle. Their especially high mobility and ranged attack allowed them to outrun virtually all other weapon systems while still firing upon them, usually avoiding harm quite successfully while still inflicting it. A good light cavalryman could even throw projectiles backwards, allowing him the unique option of retreating and then eliminating anyone who pursues him. This strategy was used extensively by central Asian peoples who raided European kingdoms occasionally, and the people of Hungary have had a strong tendency in the past to use it as well. Although heavy infantry and heavy cavalry were both heavily armored against the missiles of the light cavalry, light cavalry could still harass them into breaking formation, for instance, or otherwise foul an enemy's lines. Horses could also be shot out from under heavy cavalrymen since it was so hard to armor an entire horse.
The only weapon system against which light cavalry was particularly weak was the light infantry. Because light infantry were able to devote their full attention to throwing missiles or shooting arrows, their ability to hurt people was somewhat greater than their mounted brethren, who (especially before the introduction of the stirrup) were fairly concerned about falling off their horses. Also, light infantry were sometimes equipped with a light shield and were otherwise more free to seek cover than were cavalry, who had to remain where his horse could go. Thus, light infantry was tactically superior to light cavalry (in the sense that it was more likely to win) when the two met on an equal field.
On the defensive, the light cavalry are less useful than on the offensive, but they are still not a bad investment, although dismounting them wouldn't be a bad idea. In attacking or defending fortifications, the light cavalry was not the greatest weapon system to have, but uses could still be found for it in both places. As can be seen in the case of the Mongol conquests of the entirety of Asia under Genghis Khan, the Mongols, who were dominantly composed of light cavalry with a sprinkling of heavy cavalry, were most effective when on relatively flat land. Their favorite tactic was to flee before the enemy while shooting arrows from their extra-strong, recurved bows backwards at their pursuers. Using this tactic, they encountered great and rapid success in mainland Asia and the Middle East*. However, when deprived of their valuable mobility, as in the case of forested Novgorod, which escaped conquest, and in mountainous regions such as the Sinai peninsula, they were generally defeated.
Cavalry was generally encountered in Europe in the medeival period more than the ancient period, mostly because of the relative rarity of horses in the dominant regions of the ancient world. Neither Rome nor Greece had a whole load of horses, although there were a few in the rare plains areas of otherwise mountainous Greece. It was always, as stated before, a dominant method of warmaking among the nomadic peoples of central Asia and (to a slightly lesser extent) the Middle East. Wherever used, it was generally effective, although light infantry troops could effectively be used to counter them on the battlefield--especially if the cavalry was in forest, mountain, or otherwise out of their element.
*This certainly wasn't the only tactic the Mongols used--they had a tendency to kidnap the male populations of entire cities, driving them forward in front of the main body of the army when they approached the next city. Those people would usually be killed in the ensuing battle, which they engaged in only out of fear of death. They also had a tendency to kill everyone in a town and build towers made out of their skulls. Mongols didn't have to worry about manners.
All information herein is taken from Archer Jones's excellent and informative book, The Art of War in the Western World.