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 you with a very generic overview of how to customize your equipment when it becomes available.
Customization can be used to add “auto-abilities” to your weapons and your armor if they happen to have open/empty ability slots. The items that you obtain throughout the game can be used 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 change the overall nature of the item and may change the name of the item as well. It should be noted that while the customization may be permanent, no item or customization 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 can be sold at shops for more 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:
Finally, this page includes a list of all of the available weapon and armor customizations that you can obtain in alphabetical order: