There are several museums that you can't miss in Copenhagen; there's every kind of museum and art is everywhere: museums often host really interesting expositions.
Copenhagen (and Denmark in general) is really famous for art and design, classic and modern that spreads from the characteristic Nyhavn buildings to the most modern ones to fascinating interior design creations, furniture and components.
Here are some interesting websites.

Danish Design Center

Nationalmuseet

Carlsberg Glyptoteket

Statens Musem for Kunst - National Museum of Fine Art

Københavns Bymuseum

Museum Erotica

Louisiana Museum of Modern Art

Arken

You'll find a lot of art galleries and above all design showrooms.
You'll notice that in Copenhagen the style of designers pervades everything, from the small cafè to the big restaurant, form the single armchair to the full integration of furniture and objects.