Compliment, from Dictionary.com:
[n. kom-pluh-muhnt; v. kom-pluh-ment] Show IPA
–noun
1.
an expression of praise, commendation, or admiration: A sincere compliment boosts one's morale.
2.
a formal act or expression of civility, respect, or regard: The mayor paid him the compliment of escorting him.
3.
compliments, a courteous greeting; good wishes; regards: He sends you his compliments.
4.
Archaic . a gift; present.
–verb (used with object)
5.
to pay a compliment to: She complimented the child on his good behavior.
6.
to show kindness or regard for by a gift or other favor: He complimented us by giving a party in our honor.
7.
to congratulate; felicitate: to compliment a prince on the birth of a son.
–verb (used without object)
8.
to pay compliments.
An example of compliment used in a sentence would be:
He complimented her on how well dressed she was.
Complement, from Dictionary.com:
[n. kom-pluh-muhnt; v. kom-pluh-ment] Show IPA
–noun
1.
something that completes or makes perfect: A good wine is a complement to a good meal.
2.
the quantity or amount that completes anything: We now have a full complement of packers.
3.
either of two parts or things needed to complete the whole; counterpart.
4.
full quantity or amount; complete allowance.
An example of how complement would be used in a sentence is as follows:
The scarf she wore with her dress complemented her outfit, and set it off perfectly.
Combining the two, we could say:
He complimented her on the way in which her scarf complemented her outfit.