Bows and Arrows
An arrow is a modified version of a spear. It is smaller, lighter, typically has a set of feathers on the back end known as fletching to improve it's ballistics and a notch in the end. Arrows are used as ballistic projectiles launched using a bow. A bow at it its most simple is a cord of elastic material (such as string, sinew or silk) held between a rigid but still elastic arch of material. The arrow is notched to the string, held against the bow and pulled back, putting the string under tension and storing energy. When released, the arrow is accelerated to a high speed. Bows were soon adopted on every continent save Australia for hunting and warfare. A person trained in the use of a bow is called an archer, the skill of making use of a bow to shoot arrows at targets is known as archery.
In typical western fantasy, the Bow is the preferred weapon of Elves, which makes some sense given the facts about elves (long lives, good vision and dexterity and speed on foot), but is still a little problematic if you think about it. The biggest limitations on bow use are the physical strength and stamina of the user, who was subject to heavy fatigue, and elves have a Constitution penalty and aren't always known for lifting.
Archery in Warfare
The big advantage of a bow in pre-modern warfare was simple and obvious: range. When compared with javelins or thrown axes and spears, bows have a notable advantage in terms of rage, rate of fire and the amount of ammunition which could be carried into battle. Certain bows such as Welsh Longbows had range of up to 300 meters. Though when bows were used at such distances they were more akin to artillery than a modern sniper rifle, being loosed at the area in which enemy forces were standing rather than at individual enemy soldiers. A good archer could also loose up to ten arrows a minute, a rate of fire which is better than that of most firearms until the 19th century and the rise of breech loading rifles. The biggest downside of bows and arrows is that they are basically useless in hand to hand combat. At best, they are an inefficient club and a short brittle spear. For this reason archers in battle would carry a backup weapon in case of close quarters fighting: usualy a dagger, a short sword or a warhammer (which is also used to set anti-cavalry spikes).
As mentioned, the basics of archery were common across the world for both hunting and and as a sport. A basic bow and basic arrows are fairly easy to make and use. Military archery was less common however. Arrows loosed from a simple hunting bow may kill a deer or a naked human being, but they can be stopped by a basic wood or even wicker shield. There range was also limited. To a formation of well armored and/or shield equipped soldiers, arrow fire from common hunting bows and arrows are little more than a annoyance and would only be used in warfare as a weapon of last resort. For bows to be worthwhile against such forces, they need to go beyond a simple oak branch and hemp string. Large bows made of yew and ironwood were one option if said trees were available. The English were famous for using such bows. Another option would be to make bows by gluing together layers of horn, sinew and either wood or bamboo.
Both heavier longbows and composite bows had considerable power and could penetrate shields, mail and even plate armor at close range but were more difficult to make and were (in the case of composite bows) more sensitive to moisture. Either way, both of these types of bows took a lot of time to master, usually beginning in childhood. Archeologists can identify the skeletons of such archers in that they had very strong arms used to draw their bows. Due to their value in battle and the time which needed to be invested in training them, said archers were considered elite soldiers and were seen as valuable. In 1571 at the Battle of Lepanto, the Ottoman Fleet was dealt a major defeat. Amongst the biggest losses they had suffered was the deaths of many skilled composite bow armed archers. While their forces of galleys was reconstructed and recrewed fairly swiftly it took a decade to train replacement archers. It should be noted that at this time, both the Ottomans and the Christians were making heavy use of matchlock firearms.
Archers could fight either on foot or on horseback. Mounted Archers equipped with composite bows could be quite devastating, raining arrows on forces armed with spears and swords from a safe distance while easily avoiding the slow moving masses of men. Many ancient armed forces are known for their use of mounted archery, and the resulting rage it inspired in their foes: See the Parthians vs. the Romans, the Huns vs. Europe and the Mongols vs. Pretty Damn Well Everyone. Foot archers sacrificed this mobility for increased accuracy, increased numbers (cavalry archers also need to be trained in horsemanship and have a horse) and the ability to hide and dig in. Foot archers if used properly and backed up with pikemen, were the best pre-gunpowder counter to cavalry archers.