Quick pressure release (QPR, also called quick release or QR) means venting the steam as soon as the pressure cooking process finishes, i.e. moving the steam release handle from Sealing to Venting.
Natural pressure release (NPR, also called natural release or NR) means letting the float valve pop back down of its own accord, so basically you don’t need to do anything at the end and can keep your feet up. If it takes over 15 minutes, you're ok to release the rest of the steam (by turning the steam release handle to Venting) that may be left in order to unlock and open the lid.
Sometimes you'll see me talk of a combination of NPR followed by a QPR. For example a recipe may have a 4 minute NPR followed by a QPR. You wait for 4 minutes at the end of the pressure cooking time and then do a QPR by moving the steam release handle from Sealing to Venting.