There are two things you should eat in Copenhagen: smørrebrød and cheesecake.
Smørrebrød is a traditional lunch in Denmark. It consists of brown bread spread with salty butter and a thick topping. We had smørrebrød at the traditional Færgecafé in Christianshavn (Strandgade 50, 1401 Copenhagen). Our table was on a boat in the middle of the canal. I had a sandwich with crab and salad, which tasted really good. However, unlike the Danes, I would personally plan smørrebrød for dinner in the evening.
The Danes are also known for their love of cheesecake! So you should definitely eat a cheesecake in Copenhagen. We bought an Oreo cheesecake in Reffen... Wow! It was just sooo delicious!