How Is Compliment Defined?

A compliment is usually a positive or flattering statement made to or about someone. A compliment is usually about something to do with your appearance, such as clothes, hair etc. It's not the same as praise, which is usually given for something you've achieved. So "You have lovely eyes" is a compliment, while "That was a great presentation you gave" is praise.

A "compliment slip" is a piece of paper enclosed with a document or package; it usually just contains the sender's contact details.

A "complimentary" drink (or ticket, or whatever) is something you get for free, included in the service or maybe as an apology (eg you might get a complimentary drink if you complain of a bad meal in a restaurant.)

The link between all these meanings is the idea of politeness; in fact the word comes from the Latin complere, "to fulfil" (the requirements of courtesy.)