How do helicopters stay in the air?

Helicopters use the same principle as aeroplanes to produce lift, the main difference is how they employ it.

As with planes a helicopter uses a aerofoil to produce lift by passing air over the top of a foil faster than under it due to the geometry of the foil. The pressure difference between the air flowing fast over the top of the foil(low pressure) and slow under the foil(high pressure) results in an upwards force on the rotors of the helicopter, the pilot can essentially control the amount of force(or lift) acting up by changing the speed of the respecting flows of air. This is done by changing the angle with which the aerofoil cuts through the air.

The spinning of the rotors produces a torque, which if left unbalanced would result in the helicopter spinning like a top and not producing enough lift. This torque is countered by a tail rotor usually or a contra rotating propeller (e.g. the chinhook)