They are not effective against Bowmen because they move too slowly to approach before being killed. However, they are very effective against Villagers and later siege weapons. They tend to dominate the Tool Age for offensive capabilities but are quickly overpowered later in the game, and are best used as trash units in multiplayer games that exceed 75 population.
The increasing population and wealth of the earliest civilizations made it possible to support standing armies available at all times for defense and attacking neighbors. The first professional armies were probably built in Sumeria and Egypt. These early civilizations had much to protect and were sufficiently wealthy to provide protection. Sumerian artwork from around 2500 BC provides evidence of an early army, in this case lines of soldiers, possibly in formation, equipped with identical armor, helmets, and weapons.
—The Age of Empires manual
Axemen from all eight players
A group of Axemen in-game
Two Axemen attacking a Villager in the Definitive Edition