Here's a very good explanation: http://ramblingron.tumblr.com/post/7172299322/supplement-vs-complement-vs-compliment Basically, supplement is an addition to what is already/usually there -- such as the Sunday supplement to your regular newspaper (it's an additional magazine that is inserted into the newspaper only once a week). A complement is something that is harmonious with the thing being referred to -- something similar or of the same kind, for example, wine and cheese can be complementary (they go well paired together).

A supplement is an additional item that may be necessary. For example, people in polar regions may need to take vitamin D because they do not generate enough naturally. It is a must-have. A complement is an item that makes something better but is not necessary to have. An example would be ice cream supplemented by maple syrup. The maple syrup is not necessary but some people think that the whole combination is better.