A bladed tool, customarily used for hewing, chopping or shaping wood. Think hammer but with an edged blade in place of the flat/convex head. The back of the axe is referred to as the poll and in some axes is used as a striking surface for driving the axe further into wood using a mallet or maul. Possibly more common than hammers in the neolithic period, because flints and other breakable rocks more often break to leave edges (especially when you're using them as hammers).
Axe shapes in the medieval period depended on their use. Felling axes have very basic shapes often similar to some modern felling axes (not the American style), but others have a beard that is useful for protecting the hands when shaping boards. Early medieval axes seem to be categorised by eyes that have small lugs or none at all. Later medieval axes developed metal sheathes that extend down the handle a distance, but are not always present.
Felling axes are shaped so that the edge forms an axis of symmetry throught the centre of the axe. Broadaxes (such as the t-shaped examples show working on the ships in the Bayeux Tapestry) have this edge offset so that they can be used to shape boards and smooth planks.
The smaller, lighter throwing axe is a missile weapon, so it is not used in tournaments. Rubber and more foam is used instead of rattan in its construction (NEED TO CHECK THIS).
Historically, hand axes lost some popularity when armour got too solid for them to go through easily, but this was restored when they were put on poles (see pole axe), which gave range and increased hitting power.