How do you make a leather purse shiny?

This depends on the material it is originally made out of. If it is patent leather, you can try wiping it down with a window cleaner, such as Windex, or "polishing" it with Vaseline. If it is regular leather, you can try applying some sort of leather polish or wax. Make sure that the polish/wax is intended for purses or shoes, and avoid anything intended for saddles; it will be too harsh. If it is suede... it is not possible (although some types of suede may develop a patina/shine in areas of wear over time). As always, be sure to do a patch test in an inconspicuous area first.