Meaning of “compliment” - Learner’s Dictionary

compliment

noun uk us /ˈkɒmplɪmənt/

[ C ] something that you say or do to show praise or admiration for someone:

She was always paying him compliments.
with the compliments of sb formal

used by someone to express good wishes when they give you something free, for example in a restaurant:

Please accept this champagne with the compliments of the manager.

(Definition of “compliment noun” from the Cambridge Learner’s Dictionary © Cambridge University Press)