Axes are used for mining Melons, Glass, and many different wood items, they can also be used for PvP if you don't have a sword, and some people even prefer Axes over Swords. You can also enchant them with many different enchantments such as Breaking Speed which increases the speed that you break blocks, Momentum which makes you break blocks faster as you hold down the button, and Lumber Aura which increases the amount of Aura XP you gain for breaking Trees, or use a Moonstone Axes to break all wood type block above it.