What happens when an airplane stalls and why do pilots practice it?
Stall was an unfortunate choice of words for an engine that suddenly quits since the aerodynamic stall in aviation means something very different and isn't related to the aircraft engine at all1.  

To a non-pilot, an aerodynamic stall can best be described as the situation where there is not enough air flowing over the wings to create the amount of lift needed to hold up the airplane. 

The main reason that student pilots practice stalls is to learn the telltale signs that occur just before it happens and to make the recovery procedure automatic.  If pilots can recognize an impending stall, they can take corrective action to either avoid the stall altogether or to recover as quickly as possible.  

Outside of training, inadvertent stalls typically only occur shortly before landing and after takeoff, when the pilot gets distracted while already at a slow speed.  In both of these situations the airplane is very close to the ground, immediately requiring the correct action from the pilot in order to avoid a crash.  This needs to be instinctive and corrected using muscle memory so that it is accomplished as rapidly as possible.

The next logical question is usually:  How does a pilot fix an airplane that has stalled?  

Fortunately, airplanes are designed so that even during a stall the tail is still effective2 and the pilot is able to use it to force the nose down.  This makes the airplane go faster, since it is pointed down towards the ground, and gets more air moving over the wing which allows it to create enough lift for the airplane to start flying again.  During practice it is usually pretty uneventful, but when it happens at a low altitude there may not be enough time to regain flying speed before the airplane crashes.

For more information, AOPA has a great Safety Publication targeting flight instructors called Why we teach slow flight and stalls which is available on their website.



1 However, the sail on a sailboat can also "stall" when there isn't enough wind and since they have been around since 3,000 BC I guess that technically this usage of the word applies to both situations.  

2 There are some stalls in particular airplane designs known as deep stalls that can be unrecoverable.  I don't think that this is important when describing it to a layperson though.