The cause of dark elbows & knees is generally caused by a built up of dead skin cells from the amount of abuse these two areas get. Also from excessive exposure to the sun, heredity, extra friction on the knees and elbows, dryness, improper skin care, certain diseases, & the reaction to some types of medication.
Remedies for dark elbows and knees: Cut a lemon in half, and scrub your knees with it. Leave the juice on for as long as possible. You could also do this at night and leave the juice on overnight. Also, make sure your knees are well moisturized constantly, try to find a cream that contains green tea and cocoa butter. Scrub them gently with a body scrub or a pumice stone (if extremely dry) daily when taking bath to get rid of dead skin. Also, try not to kneel on your knees, keep your elbows off the table, & treat them with care.
Be careful to avoid commercial “skin bleachers” as these can cause very serious damage to the inner most layers of skin, including scaring that can be worse looking than the discoloration.