To neaten is to tidy up or to clean. You can neaten your room by making your bed and throwing your dirty clothes in the laundry basket.
When you turn disorder into order, you neaten. You might neaten your car by throwing away the accumulated soda cans and brushing crumbs off the seats, or neaten your outfit by straightening your tie and tucking in your shirt. Neaten comes from neat, "tidy or clean," from the Middle French net, "clear or pure." The root is the Latin nitere, "to shine."