One of the oft-overlooked museums in Copenhagen is the Designmuseum Danmark (previously known as The Danish Museum of Art & Design). I think it is a shame as the museum illustrates both Danish and international industrial and functional design history.
Housed in an old 17th century hospital the museum has a large permanent collection of historic applied arts and crafts ranging from European and Asian textiles over Chinese ceramics to colonial furniture. But more importantly there is a permanent collection on 20th century Danish design. All the famous Danish designers (Arne Jacobsen, Poul Henningsen etc) are represented with drawings, models and furniture. This is my favorite collection and it makes a visit worthwhile if you are interested in Danish design.
There’s always a number of special exhibitions on display as well. The special exhibitions differ enormously in range and focus. Currently you can see an exhibit on how Japanese design and Danish design has converged over the last 125 years, alongside an exhibit on contemporary Danish design within the entire spectrum: furniture, product design, graphic design, fashion, and design in the public space.
In the museum is a Café (complete with Danish design furniture, of course) and a gift shop.