Archery has become very popular lately, due at least in part to the "Hunger Games" books and movies, where a single well-placed arrow can bring down a rabbit, a person, or even a fighter jet. Bows and arrows go far back in human history, having long been used to hunt animals and kill people (though usually more than one arrow is needed), even if not bring down jets.
The regular bow and arrow, a long piece of supple wood with a cord, which is used to shoot a missile (arrow) far faster and harder than a human could throw it, has probably been developed several times over the millennia. Until very recently, there was nothing like the modern compound bow, where pulleys give extra power. The force of a bow was limited by one's strength, how much one could overcomes the resistance of a stiff bow--a "50 pound bow" was one that required the equivalent strength to pull back the string as picking up a fifty-pound weight one-handed. And then there's aiming. If one has ever seen a buffalo, enormous, shaggy, and fierce, then one has to admire the plains Indians trying to sneak up on them with a bow.
Bows were used in both hunting and warfare in the Middle Ages. Everyone agreed that a bow was a coward's weapon, because you could kill someone from a distance, and in practice every army marched with a contingent of bowmen. Knights would have found it shameful to fight with a bow (though they used them in hunting), but it was no use being silly and not having bowmen along.
King William Rufus (William II) of England was killed in a hunting accident in 1100. The person who killed him, who swore his whole life that it was an accident (he was given many benefices by Henry I, William's brother, who succeeded), had been hunting with the royal party, using heavy stag arrows, and the king just happened to move into range as he fired at a stag.
The late medieval stories of Robin Hood were stories of someone who was an excellent shot with a bow. But you will notice that he was an outlaw in these stories, using what would be considered a coward's weapon, not a noble lord's weapon, to fight the sheriff of Nottingham's men.
The crossbow was developed by the twelfth century. It was a shorter bow, often reinforced with metal, needing massive force to pull back the bowstring (actually a wire). Because one couldn't just draw and release, it would need to be cocked, by standing on the bow and tugging the bowstring up with both hands, until one could hook it into a notch. It would be released with a trigger mechanism. The arrows for a crossbow, called quarrels, were shorter than regular arrows. The crossbow was aimed and shot horizontally, rather than vertically. The quarrels could punch chain mail into an opponent's flesh, making them deadlier than a regular bow, but they were much slower to shoot, as reloading took a couple of minutes.
By the thirteenth century, however, a winding mechanism had been developed, so that one could crank up a crossbow much more quickly. Both crossbows and regular bows (often called longbows) continued to be used in battle into the late Middle Ages, even after the development of gunpowder. The battle of Agincourt in 1415, during the Hundred Years War, was a great triumph for the English primarily because their longbowmen mowed down the French army.