To complement is to complete something, supplement it, enhance it, or bring it to perfection. For example, your shoes may complement your dress, you and your spouse may complement each other, or minced garlic may complement a pasta dish. To compliment is to give praise. For example, if I were to say that you have a very nice turtle, this would be a compliment to both you and your turtle. Both words also work as nouns whose meanings are easily inferred from the verb senses.
A corresponding distinction applies to the adjectives complementary and complimentary. Complementary things complete, supplement, or bring to perfection. Complimentary means laudatory, and it also has a second, tangential sense: given free or as a courtesy. The coffee in the hotel lobby, for instance, is complimentary. As a verb, compliment occasionally appears in the corresponding sense to give free.
Complement has a secondary, seldom-used noun sense: a full crew of personnel, especially on a ship.
In these examples, complement means to complete, supplement, or bring to perfection, or denotes something that does this:
Too often I let academics intrude on my process, rather than complement it. [readwritepoem]
Sea trout is a beautiful fish with an earthiness that is complemented by citrus flavours. [Guardian]
The Bit and Spur’s food is a pretty sweet complement to the beer, but it ain’t on the cheap side. [Nile Guide]
Here, compliment refers to giving praise or an instance of giving praise:
A man she’d been out with three times complimented her for being “an amazing combination of fun, attractive and smart.” [Stuff.co.nz]
“A typical Rutgers-Seton Hall game,” Hill said afterward, meaning it as a compliment. [NorthJersey.com]
I say as much to Keith, complimenting Sandy on how well he managed working with a piano player with a strong left-hand. [Telegraph]
Here are a few examples of the adjectives, complementary and complimentary, put to use:
Words such as amazing, wonderful, hard-working and giving were among the many complimentary words used to describe Davidson. [Lexington Dispatch]
Dorothy Cross’s Montenotte and Fountainstown are two complementary volumes of a fascinating publication. [Irish Times]
A club lounge and complimentary glass of champagne on arrival no longer cut it for the time poor travelling business person. [The Age]
Finally, complement also bears the rarely used sense a full crew of personnel, as in this example:
The Raps have a full complement of solid starters and role players. [Rufus on Fire]