Complement and compliment are very confusing words because they are homophones. The words sound exactly alike but are spelled differently and have different meanings.
The word complement means to go together well or to round off something.
For example: The work book will complement the text-book nicely. (The work book rounds off the text-book by offering something extra to help the reader.)
For example: Meat and potatoes are perfect complements. (Of course, meat and potatoes almost always go well together.)
Compliment means to flatter or praise someone or something.
For example: The teacher complimented the student for her neat penmanship. (The teacher flattered or praised the student for the neat penmanship.)
For example: The publisher gave Piper a compliment for finishing her rough draft so soon. (The publisher praised Piper for finishing her draft before expected.)
There is no easy way to remember the difference between the two except to remember them. However, as far-fetched as it sounds, you could try to remember that complement and well (goes well) both have an e and that compliment and praise but have an i. 🙂
Tweetable: Misused Words: Complement vs Compliment