In less than an hour this morning I saw each of these two words misused. Once in a review about eggs benedict and once in another review.
Here is a piece from Grammar Girl that points out the difference:
A compliment, with an "i," is a kind or flattering remark. If a guy says he likes your dress, he’s giving you a compliment. He’s complimenting you.
A complement, with an "e," is a full crew or set, and when something complements something else, it means they go well together. You might talk about a picture frame that complements a photo or the crew complement needed to operate a ship.
A Trick to Remember the Difference
To remember the difference between the spellings of these words, be a nice person and tell yourself
I like to give compliments.
Put the emphasis on the "I" when you say or think it. The "I" can remind you that the type of flattering compliment is spelled with an "i."