Complement, as a noun, is a group that completes a set.
- A complement of four people would now bring the staff to nine.
Complement, as a verb, means to go well with something.
- The four new people would complement the rest of the staff.
Compliment, as a noun, is an expression of courtesy or praise.
- The supervisor gave the staff a compliment on its work.
Compliment, as a verb, means to praise or respect something.
- The supervisor regularly compliments the staff on its work.