Complement means to add, supplement, or complete something; compliment means to congratulate or praise politely. Thus I complimented the patissier, whose dessert beautifully complemented the meal.
In their noun forms, complement is usually a thing that completes or is a quantity of something (as in the idiomatic phrase full complement), and compliment is a kind praise. Thus I gave my compliments to the patissier, whose dessert was a perfect complement to the meal.