Abilities are skills that cost MP your character can learn. Learn them by using the appropriate skillbook. A list of abilities can be found in the Abilities Category.
Each time you use a combat ability in battle, the ability gains +20 EXP. Once you have enough EXP for that ability, it will Grade Up. A Grade Up increases the speed in which you can use the ability, as well as the effectiveness of it.
To gain the ability EXP, you must use the ability and finish casting it. The ability does not have to hit the enemy (it can miss), you will still gain the EXP for it.
Passive abilities automatically gain a certain amount of EXP after a battle.
Production abilities gain EXP as you use them.
Currently, There are five types of abilites, Approach, Long-Distance, Passive, Magical Attack and Production.
Approach Abilities are physical attacks with special effects. Because they are physical attacks, they can Miss and be Countered, as well as be Guarded against with the exception of Initial Crushing Blow.
Long-Distance Abilities also are physical attacks. They can miss but can't be countered or guarded against.
Passive Abilities process automatically with a certain probability, if you meet the requirements. For example, the passive Guard will only work if you use a shield, while Counter won't work if you use one.
Magical Attack Abilities are magic-based attacks. They can be used on any enemy, and cannot miss, be countered or guarded against.
Production abilities only work outside battle. They are used by production classes to create items or collect materials.