what does a "check valve" do? Never heard of this part of a pump.
The check valve allows you to pressurized your pump with every stroke, without having to fight the pressure you developed with your previous stroke. With a check valve, every stroke essentially starts at zero pressure. As you push the piston down, the pressure in the pump goes up. Air goes into the inner tube when the pressure in the pump body equals the pressure already in the inner tube.
You can think of the presta valve of your inner tube as a check valve. It keeps the air in your tire between stokes. The check valve in the pump body further isolates any pressurized air that might be in the space between the bottom of the pump piston and the presta valve, in the hose and pump head and such.
The schrader valve works a little differently. Since the pump heads for this kind of valve uses a pin to open the valve, the Schrader valve does not act as a check valve during inflation. This makes the check valve in the pump even more important.