The Improved Bowman is not an upgrade of the Bowman. It is a separate unit with more hit points, attack strength, and range than the Bowman. The Improved Bowman can be upgraded to the Composite Bowman.
Archers fire arrows at enemy villagers, military units, boats, and buildings within their range.
Researching Alchemy increases attack strength. Ballistics increases accuracy. Leather Armor, Scale Armor, and Chain Mail increase armor. Woodworking, Artisanship, and Craftsmanship increase range.