Crossbow
The crossbow is named for its cross-shaped design, as it is basically a bow on a stick, that uses a mechanism to trigger the release of an arrow (called a bolt or quarrel). Bolts are arrows with a shorter shaft, and could range from being lighter than an average arrow to several times heavier. It operates on the same principle as the traditional bow in that a tough bowstring is pulled back to store potential energy in the bow, which upon release of the string, transfers it to a projectile.
While historians are unsure who first made the crossbow, the earliest known crossbows were found in China around 500 BCE. During the Warring States period, crossbows became a preferred ranged weapon of Chinese armies and the use of crossbows certainly aided in the rise of the Qin Dynasty and the beginning of Imperial China. The Greeks and Romans experimented with hand-held crossbows, but they never made extensive use of them, although they did make extensive use of ballista (basically a crossbow on steroids). Crossbows began to see widespread use in Europe around 1000 CE.
The difference between a bow and a crossbow is that a bow's string needs to be pulled back and held by the user while aiming, while a crossbow has a mechanism that locks the bowstring in a readied state and only requires the user to operate the trigger to release the bolt upon sighting their target. The trigger mechanism evolved over time, as did aids for drawing the crossbow's string. The weight or size of the crossbow certainly had an effect on the mechanisms involved, as light crossbows could be reset by hand, but heavier version could end up using levers or crank-operated windlasses to pull back and cock the string, and tended to use more robust release mechanisms due to the increased stress involved.
In fantasy settings, technologically advanced races who don't (or in limitation) employ firearms as their go-to weapon typically make heavy use of crossbows. It is also the ranged weapon of choice for more richer and experienced mercenaries.
Use in warfare
The biggest advantage of crossbows over regular bows is that because they're fired using a mechanism instead of depending on human strength to hold the bow in a ready-to-fire position, user fatigue is no longer such a huge factor. Additionally, the heavier crossbows could generate more force than most humans thanks to the pulley systems used to cock the string, thus heavier bolts could be thrown, resulting in greater penetration of the target. Possibly its biggest advantage is that it was easier to train the use of a crossbow than bows since the weapon's operation is much less taxing.
Of course, the main drawback with crossbows is that they require a wider range of resources and skills to produce due to the mechanisms involved. Crossbows also generally have a lower rate of fire than bows. At best, a crossbowman can get off about two shots a minute(Now that would depend on the crossbow, wouldn't it? A weapon cocked via windlass or other crank maybe, but a weapon with a light enough draw to be bent without mechanical aid would fire much faster).
The fact that even relatively poorly trained men armed with crossbows could royally murder fully armored knights, made the crossbow one of the most hated pre-firearm weapons in the Europe, even more than the infamous flamberger. At some point it even came to the pope banning crossbows as unholy weapon, but even then they remain to be popular amongst mercenaries, rich lords and protestants. Proud knights could not accept the fact, that some dirty peasants could kill them only because their lord could afford few dozens of crossbows.
See Also: Crossbows are Underpowered in d20