A compliment is praise; but to complement something is to add something to it. Unless, of course, you're talking about a military complement which is a body of men (or, these days, even women).
Is it compliment or complement?
(Complement)
Noun:
A thing that completes or brings to perfection.
Verb:
Add to (something) in a way that enhances or improves it, make perfect.
(Compliment)
Noun:
A polite expression of praise or admiration.
Verb:
Politely congratulate or praise (someone) for something.