The best way I can describe this is free-style acrobatics. It is not properly defined as an extreme sport, nor fully a martial art, parkour encompass a great deal into its rather vague set of goals; ultimately, I think its purpose is to have fun.
Pankour stresses freedom of movement, understanding of the body and limitations, and split second reaction time. According to some, it is the art of moving over obstacles as efficiently and as fast as possible.
There is information on Wikipedia about it:
There are also several good Youtube videos that showcase it here, here, and here.
The closest thing I can relate it to in my experience is talus running; this is a technique favored by rock climbers to build up reaction time and foot speed, it involves running very fast over rocks or an uneven surface. There is also the climber concepts of the ballerina, the cat, and the monkey, that parkour seems to have adopted; these three disciplines is how a climber is supposed to face any given situation (jumping like a ballerina, landing like a cat, and using your limbs like a monkey.)
All the people in the videos above seem to use the same techniques, to an amazing affect, it is incredible what they are able to do, and the distance they are able to fall safely is simply amazing.
Like many disciplines, this one is heavy on metaphor. Proponents of parkour claim that defeating physical obstacles can help you defeat other obstacle in life; from relationships, to work, to emortional problems; they even claim that it helps your critical thinking skills. The Pankour philosophy stresses constraint, to know ones limitations and not to push too hard. Pankour experts say that, Because of this philosophy, there are very few pankour accidents, which is surprising considering what they do.
I'm skeptical that parkour has that many benefits, but as for a fun pastime that helps flexibility, reaction time, and understanding of your body (not to mention exercise), I think parkour is the next big thing.