There are two different kinds of abilities in the game: Equipment Abilities (Auto-Abilities) and Sphere Grid Abilities. Sphere Grid Abilities includes those that your characters can learn by acquiring them on the Sphere Grid such as Black Magic, White Magic, Skills and Specials. The full list of Abilities is included below.
Skills
Skill Abilities are generally a physical attack that also includes an additional added affect (Mug, Dark Attack, Power Break, etc.)
Specials
Special Abilities are miscellaneous abilities that are typically specific to one character, such as Tidus’ Flee ability or Rikku’s Use ability.
White Magic (Wht Magic)
White Magic spells are those which have curative properties and will help your party members in battle (with the exception of Holy).
Black Magic (Blk Magic)
Black Magic spells are those that deal damage to enemy combatants and are used offensively.
Most of the equipment, including armor and weapons, that drop from fiends and bosses will have a somewhat random assortment of Auto-Abilities attached to it. A full list of the Auto-Abilities that you can attach to weapons and armor is included below:
Weapons
Auto-Abilities for weapons that either come automatically attached or can be added through customization.
Armor
Auto-Abilities for armor that either come automatically attached or can be added through customization.
You can add new Auto-Abilities through weapon and armor customization. Check out the link below for additional information.