Not all characters can be customized. In most Books, it's only possible to change your main character's name and appearance at the beginning. Restarting the Book and returning to the customization screen will allow you to rename them (or choose a different skin tone, hair color or gender if available).
Some books have Closet features which can be used to change your character's hairstyle and outfits mid-scene! You can access the Closet by tapping on the clothes hanger icon next to the Settings button. If the Closet icon is greyed out and can't be selected, it means the Closet is locked in the scene you are currently on.