Now these two words are so close as to cause some confusion. They are not two spellings of the same word, but have completely different meanings.

Compliment is a verb and a noun, and refers to saying something nice about someone or their actions (pay attention to the prepositions). For example:
The young man complimented the girl on her beautiful new hairstyle. In other words he said something nice about it. By saying something nice, he had paid her a compliment.
Complimentary is an adjective meaning free of charge, so the shampoos and soaps in your hotel room are complimentary extras, because there is no separate charge for them. Buying a first class ticket on an aircraft entitles you to complimentary booze 🙂

A complement is something which adds value and improves something else. For example:
The green dress she wore really complemented her pale skin and red hair (made both look better).
Lemon juice complements the taste of fish and is often served with it.
Complementary medicine adds to existing mainstream medicine.