Unarmed Combat is considered a Weapon Skill in the sense that fist weapons have requirements for the Unarmed Combat skill. However, you can also fight entirely unarmed without any weapons at all.
The damage dealt when fighting unarmed is based on your gladiators weapons, strength and skill in unarmed combat. Thus, the skill unarmed combat is the only skill that actually increases your damage.
When fighting unarmed or with fist weapons you cannot parry an attack. This is both a blessing and a curse since it will not use up any of your gladiators actions in a round, so you will always be able to attack twice in a battle round. Read more about this in the battle basics.