Clean the area where the scratches or nicks are located using a vacuum cleaner. Ensure that all debris has been removed from the floor before continuing.
Dab a clean, soft rag with mineral spirits and rub it over the damaged area in the direction of the wood grain. Rub the rag back and forth over the area to clean any oils or small particles of dirt from the scratches or nicks.
Rub steel wool over the scratches to remove them. Again, rub in the direction of the wood grain. Apply a moderate amount of pressure but don't rub so hard that you damage the finish. Rub the steel wool slightly onto the floor surrounding the damaged area.
Apply a coat of wax using a colored wax stick that matches the shade of the wood. Apply the wax to the scratched area and the general vicinity of the damage. Ensure that you apply a coat thick enough to cover any remnants of damage that may still be present. Allow the wax to dry for a couple of hours before walking on it.