These two words are easily misused.
Complement means "to supplement" or "make complete": Their two personalities complement each other.
Compliment means "to praise or congratulate": She received a compliment on her sense of fashion.
Linguist Robert Beard, editor of alphadictionary, offers this sentence to help us remember: "Anne Chovi received many compliments for selecting vegetables that were the perfect complement to the fish for her candlelight dinner."
To complement his sentence, and indirectly compliment his work, I'll offer my own: "This blog complements your pathetic life, dear reader, so you might want to compliment me."