When compared to longbows, archery recurve bows stand out as being significantly different in shape, hence the name- recurve. Instead of a smooth arc gently curving away from the archer, the bow reverses the curve at either limb. This has the advantage of giving recurve bows an extra kick, as those limbs can pack more energy in at full draw. So a recurve bow will throw the arrow out faster, straighter, and with better accuracy.
Alternatively, for the same poundage as a straight limbed bow, a recurve bow can be made that is shorter limbed. This is a significant advantage where the bow is being used in a hunting environment, where the space available to present the bow is less.
Some modern recurve bows have the additional feature, when being used for hunting, of being easily dismantled. The limbs can be readily removed from the riser, which both helps storage, and eases maintenance. Failure of a limb doesn't necessarily mean the bow must be binned- instead only the failed limb needs to be replaced.
As for the history, the recurve bow is almost as old as mankind.
The Egyptians were known to be using recurve bows some two thousand years before the modern era. Those early recurve bows were also made from composite materials, as the modern bows are. The Romans also employed recurve bows for the Imperial Army, strengthening the tips of the limbs with lathes of wood. The development of the mounted archer was made possible, at least in part, by the recurve bow. It's short-ness enabled the ease of use when shooting from horse and it was subsequently adopted by many famous marauding nomads, including the Magyars, the Huns and the Mongols.
Recurve bows today have a basic construction quite similar, being of a laminated construction, but modern materials are generally used. So carbon foam cores, and fiberglass or carbon limbs have replaced wood and horn.
The risers can be made from wood, or plastic, in the less expensive bows. More commonly they are made from a combination of metal and carbon fiber. Overall, the weight of a modern bow can compare unfavorably with a similarly limbed long bow. But that extra mass does provide an extra stability on the release of the arrow, which leads to a greater accuracy on release, which is a critical moment.
What else is found on a modern bow? Well a molded had grip makes the hold at full draw a joy, and there is the additional aid of an adjustable bow sight, to place on the target, and nocking aids. Nocking is the process of securing the arrow to the string and the belly of the bow. A nock point and an arrow rest make’s nocking more consistent.
At the top level, in the Olympic games, the only bow permitted to compete with is the recurve bow. This is and of itself is a testament to their excellence for competition archery. Low weight carbon arrows are shot by the modern recurve bow.
These have low mass, so the energy transferred on release of the string must not be excessive. Otherwise there will be excessive vibration by the bow, which can quickly cause failure. By optimal design, archery recurve bows give those involved in today's sporting archery excellent cast, and so performance.