"Mighty Weapons are a diverse lot of one-handed clubs, axes, hammers, swords, and more, with massive size and unexpected balance as their only common traits. They are favored by the nomadic barbarians of Arreat, who possess not only the strength necessary to heft them, but the fury to swing them unceasingly in battle."— Game Guide(src)
Mighty Weapons are a special type of one-handed weapons used only by Barbarians in Diablo III. They are, essentially, heavier and more massive variants of regular swords and axes, as well as one-handed sickles. Regardless of type, their attack speed is that of axes (reflecting the primary focus on cleaving).
In addition to normal weapon stats, Mighty Weapons may also roll maximum Fury and Life per Fury spent. They can be transmogrified into normal axes, swords and maces, but only the Barbarians can do the other way.