Melee damage (or Hack Damage in Age of Mythology) is the value in Age of Empires games for how much damage a unit inflicts at close range. With the exception of attack bonuses, damage dealt by units in Age of Empires and Age of Empires II remains constant regardless of the type of unit.
In Age of Mythology, some units have special abilities that can temporarily boost the damage amount of themselves and/or other allied units around them. Age of Mythology also doesn't specifically have "Melee Damage" as all infantry use Hack Damage, including the ranged Throwing Axeman.
In Age of Empires III most ranged units have a separate melee attack. Melee attacks have different damage and different damage multipliers than a unit's normal ranged attack. If the player has advanced formations turned on for their units, they can switch a group of ranged infantry, such as Musketeers, to use melee attack. This is useful if they want their musketeers to attack cavalry, because the Musketeer's melee attack has a damage multiplier against cavalry, while the ranged attack does not.