I frequently see students confuse these two words in their writing. Many people do not even know that there are two separate spellings.
"To compliment" is to offer someone praise. As in: "Karen's boyfriend complimented her new haircut."
"To complement" is to make something more complete or perfect. As in: "Karen thinks her boyfriend really complements her life." This word is also used in geometry to refer to two angles whose degrees add up to 90 (i.e. form a right angle).