Complement can be a noun or a verb and it basically means something that completes or makes something better, or to add to something in a way that improves it or makes it better.
For example:
“This wine is a nice complement to this cheese.”
“Those colors complement each other nicely.”
Compliment is a noun or a verb, which means a remark that expresses approval, admiration or respect or to say something nice that shows your approval, admiration or respect.
For example:
“When he told me the color of my shirt complemented my eyes, I thought it was such a nice compliment!”
The adjective forms are complementary and complimentary, but be careful! Complimentary has 2 meanings: free (a complimentary breakfast) and describing a situation in which someone gives you praise (he made some very complimentary comments).
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