The metro is located right above terminal 3 at the airport. All the trains go in the same direction from the airport (M2 to Vanløse Station) and run with 4-6 minutes intervals during the day and evening. During the night the train runs every 15-20 minutes. It will take you 13 minutes to get to Nørreport Station (hub in city centre) from the airport.
The train station is located by terminal 3. You can take a free shuttle bus from terminal 1 to terminal 3, which will take 5 minutes. The trains run every 10 minutes during the day and will get you to Copenhagen Central Station in about 13 minutes. During the night the trains run 1-3 times an hour.
Bus 5A will take you directly to Copenhagen Central Station, City Hall Square, Nørreport and other stations. It takes about 30-35 minutes from the airport to the Central Station. The bus runs every 10 minutes until 01:00 at night. The buses run again from 05:00 in the morning. However, during 2013 the A-buses will start serving during the night as well.
Taxis can pick up passengers at terminal 1 and 3 and will get you to downtown Copenhagen in about 20 minutes depending on traffic. Taxis are usually waiting outside each terminal.