Complement, as a noun, means something that completes or makes perfect. It also means a quantity or amount that completes something. As a verb, complement means to complete.
Examples: 1. (as a noun) This area rug is the perfect complement to the room. 2. (as a noun) We have the full complement of encyclopedias. 3. (as a verb) A red wine would best complement our entrees.
Compliment, as a noun, is an expression of approval, praise or admiration. As a verb, it means to give someone approval, praise or admiration; to pay a compliment.
Examples: 1. (as a noun) He received a great number of compliments after the show. 2. (as a verb) My neighbor complimented me on my new haircut.