"Konfirmation" in Danish is the traditional profession of Christian faith in Denmark (and the rest of the Nordics, if I am not mistaken), usually carried out at the age of 13-15. It serves as a "confirmation" of the adolescent's baptism in the Church of Denmark when they were an infant. It is normally celebrated in April or May.
"Confirmation" in its regular meaning in English, like "Can you confirm that you are over 18 years of age?" is "en bekræftelse" in Danish ("Kan du bekræfte at du er over 18 år gammel?")
Here's more information, I don't know a whole lot about it besides the public rite in church and the party taking place afterwards. I chose not to be confirmed as I'm not religious and my parents were neutral.
http://blogs.denmark.dk/diane/2011/05/25/when-blue-monday-isnt-new-order/ - very thorough explanation http://da.wikipedia.org/wiki/Konfirmation (Danish wikipedia) http://en.wikipedia.org/wiki/Confirmation_(Lutheran_Church)
We really aren't, not in Copenhagen at least. I think it's all about tradition and very little about actual beliefs. Many people get confirmed for the sake of the money and the party, and for most of them, the biblical instruction beforehand is the closest they'll ever get to religion. I know that most of (or perhaps even all) my friends from the city area who were confirmed aren't religious. The Christian group at my high school in Copenhagen is composed of around 4% of the entire student body of 1200 students, while in Western Jutland, around half are part of those, according to a friend who lives there. Also the percentage of youth in the Copenhagen area confirmed this year fell to 40% compared to a national average of 69,8%.
Yesterday, I was at a caffe in Prague with some friends and there was a framed document from the beginning of the 20th century in German. All I could understand from it was the name Hans Sell and the event Konfirmation. I knew what that was about thanks to Duo and you!
When young people are confirmed in the state church at age 14-15, they most often have a big party with family and friends, at which they receive larger or smaller sums of money, varying in size from family to family.
But you're right; the Church of Denmark is Protestant (Evangelical Lutheran) and people who are members (everyone who is baptized in it) pay <1% of their annual income in church taxes when they turn 18. However, there is an ongoing trend of falling membership as many people opt out at age 18. Most people here aren't very religious (at least not in organized groups) and only pay church tax to preserve the churches and the culture and beauty they represent.