Why and how do airplanes fly?

Basically, airplanes fly because air is squishable (more precisely, compressible).  It's also why airplanes look a lot different from submarines -- water is not squishable.

Airplane wings are normally a bit curved, the measurement of curvature being called camber.  When a wing is moved forward, the air that travels over the top of a wing has to travel farther than the air going under the wing.  The air on top gets spread out and creates an area of lower pressure, and that pulls up on the wing creating what is called lift.

Lift can be changed a few ways.  If wing camber is increased (it gets more curved), we get more lift -- to a limit.  Or if we make the wing travel forward faster (increase airspeed) we'll also get more lift.  If the wing is in a more dense gas (sea level instead of 10,000 feet altitude), there will be more lift as well.  And if we rotate the wing front edge up with respect to the air (known as increasing the AOA, angle of attack), we'll get more lift -- to a limit known as stall.

Play around with the following wing and see how these factors change the amount of lift from a single wing.
WING EXERCISE

Lift is not the only thing generated by a wing -- the passing air also pulls back on the the wing, generating a force known as drag.  In general, the more lift a wing produces, the more drag you get as well.

To fly, an aircraft must keep its wings going through the air fast enough to support its weight.  In other words, the lift from the wings must equal the weight of the aircraft to fly straight and level in the air.  To stay at a certain speed in the air, the thrust of the airplane must equal the drag on the airplane.

Wings aren't the only important parts of an airplane -- there are also things called control surfaces which allow a pilot to point the airplane in the direction he/she wants to go.
AIRCRAFT SCHEMATIC

The horizontal tail is pretty much a small wing at the back of the airplane -- it acts just like a wing and creates lift.  The elevator is a small movable piece at the back of the horizontal tail that moves the same way on both sides -- changing the position of the elevator changes the lift produced by the horizontal tail.  Since it's so far away from the wing, and wing is usually near the center of the airplane, pushing up with the horizontal tail causes the nose of the airplane to go down (also called pitch down) and the airplane to dive.  Conversely, pulling down with the horizontal tail causes the airplane to pitch up and climb (provided the airplane has enough thrust).  Sometimes the horizontal tail is built so that the whole thing can move -- if this is the case, it's called a stabilator or an elevon; if  the stabilators can move opposite one another, this will help to roll the aircraft (see ailerons).

The vertical tail is a half-wing at the top of the back of the airplane -- it acts similar to the back of a dart and keeps the airplane nose forward.  A small piece of the vertical tail called the rudder can be moved and causes the airplane to twist (called yaw) left or right.

A small piece on each wing called the aileron changes the effective AOA of the wing -- these pieces move opposite one another, so that if the right aileron goes up the left aileron goes down.  In this case, the AOA of the right wing goes up creating more lift, while the AOA of the left wing decreases causing less lift -- the airplane will rotate (or roll) to the right.

The aircraft pilot has controls to move each of these surfaces.  The rudder is controlled by the pilot's feet on the rudder pedals -- pushing with the right foot causes the rudder to deflect and point the aircraft in that direction.  The elevator is controlled by forward-aft motion of the stick or yoke -- pushing forward causes the elevator to deflect and point the aircraft down.  The ailerons are controlled by left-right motion of the stick or left-right rotation of the yoke -- right motion on the stick causes the ailerons to deflect and rotate the aircraft in that direction.
 

Why do some airplanes not fly like normal airplanes?

HARRIER PICTURE

Sometimes, airplanes can do some special things.  Take, for example, the AV-8B Harrier -- it has the ability to provide lift with its engine as well as its wings.  The same rules apply to fly (lift equals weight, and thrust equals drag) but the force of the engine can be directed downward in this case.  This allows the Harrier to take off at very slow forward speeds or even straight up by pointing the exhaust of the engine down, then rotate the exhaust backward until it flies just like a normal airplane.  With the Harrier, you have to specify whether the airplane is in engine-borne flight (aircraft lift from the engine), wing-borne flight (lift from the wings), or somewhere in between.
 

So how do helicopters work?

HELICOPTER PICTURE

Helicopters, also called rotary-wing aircraft, still have to obey the same laws of physics, but they do it in a little different way than fixed-wing aircraft.

The main rotor blades of a helicopter spin fast enough that each one acts like a wing, generating lift. The angle of attack of each rotor blade can be changed. If all blades are changed in the same manner, this affects the overall lift on the aircraft causing it to rise or fall -- pilots control this by moving a control device called the collective located to the side of their seat up and down. Also located on the collective is the throttle control which is twisted just like the throttle on a motorcycle. Just like in a fixed-wing aircraft, faster airflow means more lift so speeding up the engine causes the rotor blades to turn faster and generate more lift.

Helicopters don't have ailerons or elevators, but they do have a way to change the angle of attack of the rotor blades whenever they are over a certain direction -- the control stick on a helicopter is how the pilot does this, so whichever way he pushes on the stick is the direction that gets less lift and causes the helicopter to go that way. Just like in a fixed-wing aircraft, this controls both pitch and roll.

Helicopters don't have a rudder, either, but instead they generally have a smaller set of rotor blades turning along a boom to the back of the aircraft. The tail rotor generate lift in the sideways direction. Without a tail rotor, a helicopter body would helplessly spin in the opposite direction from the main rotor. With the tail rotor spinning at the right speed, this doesn't happen -- in fact, there is generally a link with the throttle so that the tail rotor speeds up just enough when the throttle is increased so that the helicopter still will fly straight. The foot pedals on a helicopter control the speed of the tail rotor so that they work just like a rudder to control the yaw of the aircraft.