As mentioned by m1a9366b's rather extensive write up, chicken is a game, if you will, played by consenting drivers whereby two cars race toward each other at high speed, willing the other to swerve first. It is very important to know which way you should turn, otherwise you may both look rather the worse for wear. Always swerve in the direction that you country's road rules follow. That is, in Australia and England you should turn left and in America and almost everywhere else you should go right. Observe by this ascii art:

            _____|   | |
___________/     |___|_|
           _____   _________        In England/Australia
          | |   |_/         

           | |   |___
           |_|___|   \______
_____     _____                     Everywhere else
     \___|   | |

You would look pretty stupid if this happened (you wouldn't however, look pretty)

_____     _____ * _____      ______
     \___|   | |*| |   |____/

Excuse my ascii - It is my first attempt ever.
Chicken is a popular stunt in the movies, such as Death Race 2000, Cry Baby and Mad Max

In High School High, there is a different kind of chicken played, called a Chickie Run, where two cars race toward a cliff, the first person to jump out being the loser (kind of a cross between car chicken and train chicken). sid tells me that the same thing happened in Rebel Without A Cause - the High School High name is probably a reference.