What sort of things are you interested in?

Options include:
Shopping and street scene (walk along Strøget);
Museums and churches (Ny Carlsberg Glyptotek, Marmorkirken, National Museum, etc);
Modern art (Louisiana)
Carnival and funfair (Tivoli, has dining options)
Smaller towns (Train to Roskilde)
Smaller towns and castle (Helsingør and Kronborg Slot)

If you really have no idea what you want to do, here's an answer:

Buy a transit day ticket.
Take the metro from the airport to Kongens Nytorv.
Walk south-west along Strøget (longest pedestrian shopping arcade in the world), put your nose in the Phaidon shop, keep going and look in more shops if you want, get to Illum and go inside to buy some Danish Design housewares to take home.
Fortify yourself with a hotdog or other sausage from one of the sausage selling vans along the way, and then an icecream.
Optional: Divert to your left a little to look at Charlottenburg Castle, Folketinget (Parliament building, etc), the National (Royal) Library with the startling new section "The Black Diamond", then return to Strøget to carry on.
From Strøget, at the old square ("Gammeltorvet") look at the Copenhagen Court House with the inscription "Med lov skal man land bygge" (With Law Shall [a] Land be Built).
Carry on to Rådhuset (City Hall) square, go inside to view Hans Olsen's World Clock.
Cross the main road by the Danish Industries House (large glass fronted building covered in logos of Denmark's more significant manufacturing and industrial companies) and go on to Tivoli.
Enter Tivoli, look around, have dinner. Tivoli is open late, unlike most of the rest of Denmark.
Exit Tivoli to the west side, go to the central station (Hovedbanegard) and find the train to the airport (usualy around platform 5-6, it's the train to Øresund and Malmö which stops at the airport).