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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.
Mounted archers were once the most efficient part of the army. Many civilizations owed to them their successful conquests. Since firearms replaced bow and arrow they no longer fight but they participate in an attractive sport. Read about mounted archery.
Japanese archery is an old tradition which is perfected for centuries. It appeared as the military class took the power over the country and it lasted until firearms appeared and became more effective.