complement / compliment are similar-sounding terms with different meanings (referred to as homophones). To better understand the differences, see below for definitions, pronunciation guides, and example sentences using each term. 👇
complement: (noun) a word or phrase used to complete a grammatical construction. (noun) a complete number or quantity. (noun) number needed to make up a whole force. (noun) something added to complete or embellish or make perfect. (noun) one of a series of enzymes in the blood serum that are part of the immune response. (noun) either of two parts that mutually complete each other. (verb) make complete or perfect; supply what is wanting or form the complement to.
compliment: (noun) a remark (or act) expressing praise and admiration. (verb) say something to someone that expresses praise. (verb) express respect or esteem for.
complement: k·AA·m·p·l·ah·m·ah·n·t
compliment: k·AA·m·p·l·ah·m·eh·n·t
(Examples are Wikipedia snippets under the CC ShareAlike 3.0 license. Definitions drawn from WordNet.)