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Simple answer: By blowing air at the ground.

Long answer: Air moving in the right way can have immense force behind it. Hurricanes can flatten trees, tornadoes can pick up cows and jet aircraft can use a directed jet of air to propel itself.

The Rotor blades of a helicopters are like a massive, high powered fan that blows air downwards. Each rotor blade has an airfoil profile (similar shape to the cross section of an airplanes wing), so as the rotor blade turns it generates lift, just like an airplanes wing. The when the blades turn fast enough the lift forces will be greater than the forces of gravity. The pilot can control this by altering the angle of the blades. As the pilot increases the angle of the blades, more lift will be generated. In practice all helicopters do not have much variation in rotor speed during flight, and the rotor speeds are a lot slower than most people realise.

But there's a problem known as Torque Reaction. If you just have a single main rotor, then as the helicopters blades starts to get to a state where lift from the blades equal the forces of gravity, the helicopter will start to spin on the spot. (A visual thing to see this in action is watch the film Black Hawk Down, when the helicopter gets shot and loses its tail rotor).

To balance the forces induced by torque reaction, most helicopters have a tail rotor at the end of the tail boom. The tail rotor blades have a variable angle controlled by the pilot, like the main rotors. Adjusting the angle of blades will balance the forces of torque reaction. Altering this angle in a hover will allow the helicopter to turn on the spot.

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