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During the long history of China, archery held a very important place in the life of its citizens. It was one of the Six Noble Arts and a method of choosing officers in the army, a method of hunting and an art itself. Read more about history of Chinese archery.
Crossbow is a younger sibling of a bow and arrow. It is easier to use and more precise with less effort but slower. That is why it replaced bow and arrow in battlefields until it was itself replaced with firearms.
Compound bow is one of the most modern designs of a bow. It uses more complex physics to give more power to the arrow and allows an archer to hold the arrow in full draw without too much strength.
Composite bow, although a bow if a pretty complex design, is old weapon and many civilizations knew about it and used it above others. It can be made to be much smaller and stronger but it could last for a shorter time if exposed to humidity.