The palm mute, also known as palm muting, is executed by resting the edge of the picking hand's palm on the strings near the bridge, producing a dampened sound. The name is a slight misnomer, as muting is performed by the heel of the hand in general.
Half the battle to playing technically is not playing anything else but what you want. When trying to stop extra noise on the strings above the pick, use the palm of your picking hand. When you want the strings to be quiet below the pick, use the fingers of the fretting hand that you are not using.