There are two different types of skills. Combat skills and Job skills. Both are leveled up when using abilities connected to that specific skill.

When using an Ability the user gains skill points (SP). If enough skill points have been earned the skill will level up. A higher level in a skill will affect the effect of the abilities from that skill and will unlock new abilities that can be bought or found.

But if you were a mage, and you attack with fire staff, you would level up fire magic skill to do more damage, same goes for ice and shock staff mages can level up different magic depends on which staff you use.