Sometimes abbreviated as 'MeD', melee damage is, as its name suggests, the amount of damage dealt on a successful melee attack. Melee damage is reduced by the protective value of the target.
An attack has a base melee damage depending on the weapon (unarmed strikes deal 1d2, 1d3, 1d6 damage for small, medium, large creatures respectively - warrior monk/nuns do more), this amount is modified by strength (the modifier is simply added to damage), rings of damage (increase damage based on the ring's modifier), weapon skills with the weapon you are wielding, and tactics. If you are wielding two weapons and the left hand weapon would normally deal xdy+z damage, z will be reduced by 50% (this penalty can by mitigated with skill in fighting with two weapons).