How does a propeller fly?

How does a propeller fly?
The blades of propellers are angled (and shaped) to push large volumes of air back at high speeds.
In an airplane, the air pushed by the propellers moves the plane forward by Newton's 3rd law (any force applied to an object causes an equal and opposite reaction - here the air goes back and the plane moves forward). The plane flies because the shape of the wings gives lift when they are pushed through the air at sufficient speed.
In a helicopter, the air pushed by the propeller lifts the plane from the ground (still Newton's 3rd law). Helicopters require much bigger engines and propellers (per pound) than airplanes because the lift of wings is much more efficient than the brute force of the propeller. When helicopters move forward (backwards, sideways, etc.) it's because the propeller is tilted slightly causing some of the force to be applied backwards - which pushes the chopper forward.