Compliment (n):
1. an expression of praise, commendation, or admiration: A sincere compliment boosts one's morale.
(v) 1. to pay a compliment to.
2. to show kindness or regard for by a gift or other favor: He complimented us by giving a party in our honor.

Complement (n):
1. something that completes or makes perfect.
2. the quantity or amount that completes something.
3. either of two parts or things needed to complete the whole; counterpart.
(v): 1. to complete; form a complement to.
-from Dictionary.com
 

Compliment and complement can be rather tricky considering the only difference in their spelling is the second vowel. Once upon a time, they could be loosely substituted for each other. However, their more closely related definitions have become obsolete. Now, one letter makes a big difference in meaning.

Compliment deals with the giving of praise or the act of giving praise. It can be used as a verb (He complimented her) or a noun (He showered her with compliments). Its multiple definitions are due to the varying purposes of praise:

Ex. The governor gave his guests the compliment of a personal tour of the capitol building. 

Here compliment refers to showing honor.

Ex. Shane complimented the artist on the piece; it truly was the best work he’d seen.

Here complimented means giving praise and implies admiration.

Complement deals with the achievement or act of achieving completion or perfection. It can be used as a verb (Yin complements yang) or a noun (Flowers were the ideal complement to the centerpiece).  Complement’s definition varies depending on the field being discussed, but always refers to a part that allows a whole to be achieved.

Ex. Coffee is the perfect complement to any morning.

Ex. Their voices complement each other’s.

Ex. A 30° angle complements a 60° angle because it provides a sum of 90°.

Ex. A scarf would be the ideal complement to this outfit.

So if you want your work to be complete, perfect and warranting praise, be careful about that one letter.