I have a polygon which belongs to a layer and I need to move it to another one. Is there a way to do it without drawing it again?
Yup, edit the layer you want the polygon from by clicking on the 'pencil' icon. Then select Copy features icon (in the red circle shown) or from the Edit Toolbar. Select the layer you want the polygon to be moved to and select the Paste features icon (or from Edit menu).
To move the polygon, select Move features shown in the blue circle or (you guessed it!) from the Edit Toolbar.
Hope this helps!