The metro is located at the end of the airport arrival hall and will take you to Kongens Nytorv or Nørreport in Copenhagen city centre every four minutes. The journey takes less than 15 minutes.
The regular train is located on the lower level of the airport arrival hall and will take you to Copenhagen Central, Nørreport or Østerport in less than 15 minutes.
More details in tickets, zones, etc in this guide on getting from the airport to Copenhagen city center.