Copenhagen is to a large part characterized by it's many canals and the harbour (well, and the little mermaid of course). It makes sense therefore that one of the easiest (and prettiest) ways to explore the city is by boat. There are a couple of companies offering boat sightseeing trips - all of them situated at Nyhavn.
I did the tour twice and would do it again immediately. Both times I chose the company Stromma and although it's the most expensive tour you can get in the city, I don't regret spending the money. You're out and about for an hour and have a guide on board who explains everything in Danish, English and German (at least).
Starting in Nyhavn, you first make the way out to the Opera House, the harbour, past Amalienborg Castle and on to the little mermaid (who was on world exhibition in Asia the first time I visited Copenhagen so I was glad to see her for the first time last summer) before going on to Christiania and the canals of the inner city.
I wouldn't recommend boat sightseeing anywhere - Stockholm can be explored perfectly fine on foot, same as Gothenburg even though I enjoyed the boat trips in these cities - but it's definitely a must if you're in Copenhagen!
Have you ever been on a boat sightseeing tour? Where?