Weapon and Armor Customization
You gain the ability to customize your weapons and armor after Rikku joins your team and after you reach the town of Guadosalam. The game will provide a generic overview of how to customize your equipment when it becomes available.
You can use customization to add “auto-abilities” to your weapons and your armor if they happen to have open/empty ability slots. You can use the items that you obtain throughout the game as ingredients to add customizations, with more powerful and rare items enabling you to add much more powerful customizations to your equipped items.

Adding customizations may alter the overall nature of the item and change its name as well. Customization is permanent, but there is no item or customization that is so rare that you couldn’t purchase or obtain the raw materials to customize it again, so don’t worry about making mistakes.

Weapons and armor with empty ability slots tend to be more valuable, and you can sell them at shops for a higher price in gil. You can purchase empty three and four-slot weapons and armor at some of the shops around town as well. Check out the “Best Armor” section for more details:
Best Armor

Details on how to craft the strongest armor in the game, including a complete list of the strongest auto-abilities.
Finally, this page includes a list of all the available weapon and armor customizations that you can obtain in alphabetical order: