A photo of a military helicopter.
Helicopters fly because of the interaction between their moving parts and the surrounding air. A helicopter has a main rotor on top that spins, and this rotor has blades attached to it. As these blades spin, they push the air down, creating what is called lift. This lift, in turn, pushes the helicopter up into the air.
The way the blades work is pretty cool. Each blade acts like a small wing, and since the rotor is spinning, the airspeed over the top of the blade is higher than underneath. This difference in speed creates lower pressure on top and higher pressure below, so the blades lift up, which lifts the whole helicopter up into the air. There is more than just lift, however. The pilot can control how much lift is created by changing the angle of the blades, also known as the pitch. Therefore, if the pitch goes up, the blades will push down more air, and the helicopter goes up. If the pitch goes down, it means that there's less lift, and the helicopter either goes down or just stays where it is.
Now, here's where it gets a bit tricky. Helicopters also have a tail rotor. The main rotor spinning would make the helicopter want to spin in the opposite direction, so to stop this spinning, there's a smaller rotor at the back (the tail rotor). It spins and pushes air to the side, controlling the helicopter's direction and keeping it from spinning around uncontrollably. Adjusting the tail rotor's power also lets the pilot turn the helicopter left or right.
To move forward, backward, or sideways, the helicopter uses something called cyclic control. With cyclic control, the helicopter changes the pitch of the blades at different points in their rotation. So, if the pitch is higher in one part of the rotation, the helicopter will tilt in that direction, moving the whole aircraft. Meanwhile, to go up and down, there's collective control. It works by changing the pitch of all the blades at the same time, which reduces or increases the amount of lift produced. If you want to go up, you increase the pitch; if you want to go down, you decrease it.
The power for all this spinning comes from an engine. Most helicopters have one or two engines driving the rotors through a gearbox. The gearbox ensures the rotors spin at the right speed because typically, the engine spins much faster than the rotors need to be. It's this connection that allows helicopters to fly, but it also requires everything to work perfectly together. Even when one thing isn't pulling its weight, everything falls apart.
Now, flying a helicopter isn't exactly just a question of buttons pressed or levers pulled; rather, it is more of how the helicopter responds to such controls. For a pilot, it is necessary to feel just how each little adjustment will change that flight, whether it be hovering over one place, rising, falling, or forging ahead. All these are controls which the pilot must balance all at once, which is why helicopter flying needs a lot of practice.
The way they land or take off is unique to helicopters. Unlike airplanes, they are able to take off straight up or land straight down. This happens because of all the control over lift and direction, as discussed above. To take off, the pilot increases the collective to make more lift, and once the helicopter is off the ground, he can adjust the cyclic to move or hover. In order to land, everything that is done for taking off is done in reverse and in an opposite way.
In a nutshell, helicopters fly by spinning their rotors to push air down, creating lift. The pitch of the blades can be changed for more or less lift, the cyclic control tilts the helicopter in one direction or another, and the tail rotor keeps everything from spinning out of control.