This is a confusing pair of words for many, with complement often being used when a person means compliment. As the former is less frequently used in general, I don’t think they are often confused in the other direction. 

A compliment is a kind comment or well wishes. For example: ‘I love your dress!’ or ‘You have a beautiful singing voice.’ or ‘You are such a kind person.’ 

Complement has a number of different uses. Per dictionary.com, it can mean ‘something that completes or makes perfect.’ For example: This cheese complements the burger. 

Another meaning is ‘the quantity or amount that completes anything.’ It can be objects or people, such as: We have a full complement of players for the game.

In grammar, a complement is a word or phrase in the predicate that completes a particular construction – small in: The ball is small; on the cart in: She put the book on the cart. Complex grammar, yay!

Complement is even a math term. Two angles that equal 90 degrees are called complementary.

For music, complementary intervals put together equal an octave.

Finally, in art, complementary  colors are pairs of primary and secondary colors that work well together, such as red and green or yellow and purple.

As you can see, compliment and complement are only different by one letter. That little vowel makes all the difference. A way to help you remember which one to use as a kind word to someone is ‘I give a compliment to you’. 

I will give you all the compliment of believing that you will know the difference from now on and use the correct spelling for these two words. Go forth and grammar well!