Another in a series of words that often get mistaken.
"The carrot to compliment this stick is that ..." - Big Issue article on ID cards, March 17-23, 2008
The carrot actually complements the stick.
And a new example with slightly different usage just in on 14/04:
"If you’re a fan of Neue Helvetica, we invite you to take a look at Nimbus Sans Novus™. ... The families are large and complete. However, the Nimbus Sans Novus character sets include a compliment of Eastern European characters." - FontHaus mailing
Here, the set includes a complement of Eastern European characters.
A compliment is praise - and complimentary describes something nice said or written about something or someone. (Or something given free - for some reason I don't really understand. "A complimentary cup of coffee" presumably says nice things about you whilst you are drinking it!)
To complement is to add to, or to make something whole or more complete. Complementary describes something adding to, completing, or going well with, another.
LissaT
Pro 
That reminds me of that joke about the man in the pub who kept hearing voices when he is at the bar so he decides he needs a cigarette to calm his nerves and then starts hearing a much less friendly voice. The barman tells him 'the peanuts are complimentary, sir, but the cigarette machine is bang out of order'.
The one I always have to check is discrete and discreet.