The palm mute (known as pizzicato in Classical guitar terminology) is performed with the picking hand. While picking, the side of the palm is lightly pressed against the strings to mute the notes. You’ll see it labeled in tablature with “P.M.”.
You can vary the sound of a palm mute by adjusting the applied pressure. For example, light muting can be done with a light touch and heavy muting with a heavy touch. Also, palm muting near the bridge will produce a lighter muting, whereas a heavier muting can be produced by palm muting further from the bridge.