'Abilities' are the different abilities that your character can use during or outside of battles. They also decrease your MP if you use them, except some of the Support Abilities. Abilities can be learned from the equipment that the character is equipped with. Each of these abilities requires a certain number of AP or Ability Points in order to be learned.
Learning abilities maybe quite confusing and tedious. Once you've equipped a new weapon, armor, etc., you can actually use the ability/ies that the equipment have. But, if you remove the equipment while the learning process is still incomplete, you can't use the ability anymore. If you have completed the learning process (completed the required AP), you can still use that ability if you remove the equipment.
APs or Ability points can be acquired by fighting enemies. You won't receive any if you run away though. Now, to be able to make the ability acquire APs, you must equip it first in the ability sub-menu. Equipping Abilities also need Magic Stones which can be seen in the character frame (character's picture and his/her information). It presented in a fraction. I.e. 8/10 you can use 8 magic stones and you can have a maximum of 10. As you level up your character, the number of Magic Stones will be increased so don't worry.
Now, if an ability needs 40 AP, you must acquire a total of 40 AP from battles. After accomplishing these, you can now use that ability. Take note also that NOT all abilities can be learn by a character. There are certain abilities that can only be learned by a certain character. It will be grayed- out if it cannot be learned (another clue is that it has no AP requirement).
There are also 2 types of Abilities, the Support and the Attack abilities. The attack abilities are those that you can use during battles against your opponents or cure your characters. Support abilities, on the other hand, helps your characters by boosting your fighting ability (+Strength, +speed, etc.).
If you're familiar with the Junction System in Final Fantasy VIII, you'll notice that the Abilities (especially the support abilities) in FF9 has a resemblance to each other.
Status Abilities: Some weapons contain additional status abilities such as Silence. You'll need to equip the Add Status Ability. Example, let's say you have a sword with the Silence added ability. If you equip the Add Status Ability, when you attack monsters using the weapon, it will induce the Silence condition on the enemy.