I would like to give a compliment to you. Perhaps it’s your fabulous grammar know-how. Perhaps it’s your bravery to tackle the sometimes nonsensical English language head-on. Perhaps it’s your awesome earrings. (Where did you get them? Can I borrow them sometime?)
Whatever the reason, remember my first sentence: “I would like to give a compliment.”
If you are looking for the word meaning an expression of praise or admiration, you’re looking for a word with an “i” in it: compliment.
If you are looking for the word meaning something that completes, balances, or pairs well with something else, you’re looking for the form with an “e”: complement.
So if I just complimented you on your earrings, I could note that they complement the necklace that you’re wearing. Or perhaps the sparkle of sun on the Venice canals. Either way, you get the idea.