The unarmed skill is an innate skill; all classes knows it from the start, hence you do not need to visit a Weapon Trainer to learn it.
The unarmed skill is a requirement for wielding and using Fist Weapons (such as the Dagger skill is a requirement for daggers).
If you hit a foe whilst..
a) Being disarmed.
b) Having no weapons equipped.
c) Having Fist Weapons equipped.
.. you will be using your unarmed skill - hitting things with your fists.