Axes have higher damage than Swords but they have penalty to Attack and Defence, making them imprecise, hard to defend with and somewhat unpopular among players. Pick Axe is a rarer variant of axe and it has horrible attack and defence penalties but high damage.
Battleaxe is a well-rounded two hand axe with high damage and good length. Unlike other Axes it doesn't get any penalties.
These are some of the magic axes that are found on units or that can be forged.