Air is a fluid. Fluids, as they accelerate, lose pressure. If you notice the top surface of a wing, in cross section, is more rounded than the underside (in most designs). This curvature causes air to accelerate over the top of the wing, which causes a low-pressure zone to form over it along the wing's contour line (and other parts of the airframe too, as those surfaces are similarly curved and contoured). This low pressure acts like a suction, pulling the aircraft up into the air against gravity. This is what we call "lift". If the airflow over the wings is sufficiently interrupted by, for instance, a very nose-high angle, the air flow over the top of the wings gets interrupted. In other words, it doesn't flow against the surface of the wing as much, or even at all, depending on how high an angle difference there is between the direction of the flow of the air, and the angle of the plane. Since more air is hitting the bottom of the wing in a nose-high angle, and very little, if any over the top of the wing, this "lift" force goes away, and the weight of the plane is acted upon by gravity. The plane then drops, usually with a nose-down angle, in a free-fall basically towards the ground.
This loss of aerodynamic lift is called a "stall". Unlike a stall in a car, the airplane engine keeps running (unless it's a jet where one can actually have a compressor "stall" in addition to a wing stall, but that's another explanation).
The way one breaks out of a stall is by reducing power, pointing the nose down even further (yes, this sounds counter-intuitive and nuts... but it's how you do it), and then you bring the nose back up carefully, and gradually, while putting power (throttle) back in. It can actually be fun to practice these, as it's almost like a roller coaster ride. With practice, one can actually lose very little altitude before a recovery is fully realized. At the beginning though, one tends to lose quite a bit of altitude during initial training.
The reason we do training for stalls is for the sake of recognition of the stall condition during takeoffs and landings, when the plane is low, and slow, and loss of altitude is the most dangerous. Quick recovery from such a condition is essential to continued survival.
Not all planes stall out "violently" with the nose heading to the ground either, so it's important to practice in each model you fly. A Cessna 152, or 172, etc., stall nose down. A Piper Warrior, or Archer though, tend to stall "flat" and all you may feel is some vibration while the stall warning horn goes off.
A way you can experiment with stalls without being a pilot is to drive with your arm out the window (carefully), and with your arm held straight, and your palm parallel to the ground, you'll feel your arm want to rise. This is caused by lift being generated. Turn your palm so it is at more and more of an angle to the ground, until it is almost perpendicular. You'll notice that as you do so, your arm wants to drop or is pushed backwards. This is due to the loss of lift. Same thing that is happening to the airplane's wing.
This lifting action, by the way, works on the same principal as carburetors, and vacuum cleaners. Air accelerated into the vacuum cleaner's hose loses pressure due to the acceleration, and therefore causes a "suction". Pretty neat huh? This is also known as the Bernoulli principal (fluids losing pressure as acceleration increases).
Hope this is of help, and hasn't completely confused things.