Add a background map for example the OSM standard map.
Go to Layer → Create Layer → New Shapefile Layer …
Set the filename (and path), choose the geometry type (in this case; polygon) and the projection. In most of Denmark use EPGS: 25832.
Right click the shapefile and choose Toggle Editing.
Press the Add Polygon in the upper menu and start drawing your polygon on the map. Right click to end the polygon editing. Press OK in the Feature Attributes.
You have now created a polygon as a shapefile. Right click the shapefile and press Toggle Editing to end the editing session.