The words complement and compliment can be used as verbs or as nouns. The pronunciation is the same for both.
As a noun, complement most commonly means “something that completes something else” or improves upon it:
Dessert was the perfect complement to the meal.
Complement also refers to “the usual number or quantity” of a thing:
The ship departed with a full complement of cargo.
When used as a verb, complement is similar to the first meaning above:
Dessert complemented the meal.
(Complement is also a grammatical term and a medical term.)
The noun compliment refers to an expression of praise, admiration, respect or regard, sometimes accompanied by a gift:
“My boss paid me the nicest compliment today.”
(plural) “Please enjoy this free appetizer, compliments of the chef.”
The verb compliment is used to express praise or approval, or to show regard or respect:
“I’m going backstage to compliment Marlo on a fine performance.”
He complimented her on her good taste.
The adjectival forms are complementary and complimentary:
The two positions are complementary; both roles are indispensable.
This complimentary pass gives you full access to our services for one month.
Complementary can also be used to refer to colors and to medicine. “Complementary function” is a mathematical term.