One could argue that most writers confuse these terms as an error of spelling rather than usage—all the more reason to distinguish between them carefully and avoid an embarrassing, sloppy habit. The distinction is simple: “complement” (note the “e” in the middle) means “something that completes”; “compliment” (note the “i” in the middle) means “to express praise” or “thanks to”:
Jupiter Scientific Publishing Company recently published The Bible According to Einstein: a Scientific Complement to the Holy Bible for the Third Millennium.
The compliments provided in the “Acknowledgments” section of this manual are provided compliments of the author.
Spelling “complement” correctly is especially important in fields such as biochemistry, where “complement components” and “complement pathways” are cited frequently.