Valentine's Day was never traditionally celebrated in Denmark, so no, we don't exchange gifts or cards on this day.
Like Halloween, it has made its way in - mainly amongst the young, due to all the American films and TV series shown.
In Denmark, you traditionally (if you were unmarried) sent a card that you had stuck snowdrops into and wrote a number of dots corresponding to the number of letters in your name and a little verse, which would sound rude in translation, but isn't in the original, so I shan't attempt a tranlation.
The recipient had until Easter to guess who the card was from. If they guessed correctly, you gave them an Easter egg on Easter Sunday. If they could not guess who had sent the card, you claimed an Easter egg from them.
Sadly the tradition has more or less died out here, being only remembered by children in kindergarten or day-care and their grandparents.
Originally, like Valentine's Day cards it was a way of telling the person you were in love with that he was welcome to make some kind of advance.