I sometimes get these two words confused so I thought I would share with you what I have found.
Complement comes from a Latin word that means "to complete". If somethings complements something else then it enhances it, makes it perfect or completes it.
A wine can complement a meal, or a shirt can complement someone's eyes.
Complementary colors are colors that are directly across from each other on the color wheel and enhance each other because of their contrast such as red and green, purple and yellow and blue and orange.
Compliment on the other hand means "an expression of praise, commendation or admiration". As a verb it can be used to "give someone praise or say nice things" about them.
You can compliment someone on a job well done, sometimes people give "compliments to the chef" after enjoying a nice meal.
You an pay someone a compliment by telling them that they look nice today or they did well on something.
Something that is complimentary is something that is free or given at no charge. On international flights airlines serve complimentary drinks to passengers.
I hope this helps!