You can craft equipment, items and abilities from the Craft menu found on the Home screen.

Recipes, materials and Gil are required when crafting.
You can obtain recipes while playing through the main story or as event rewards.
The materials needed to craft an item can be dropped by monsters, found in chests throughout exploration maps or from limited-time events.

With the recent update, items that are ready to be crafted can be made instantly as long as you have sufficient materials.