Sport originally derived from sport climbing, consisting in short sequences of "moves" (movements of the arms allowing the body to be moved up a climbing wall, a rock boulder or even a house wall) that requires huge levels of strenght on the forearms & fingers, and different special tecniques. This sequence is normally called a "problem". The climber is not roped, and the only protection allowed is a thick foam mattress called "crashpad". Bouldering problems rarely go higher than 3 or 4 meters above the ground.
"Solving" a problems often forces to make unlikely movements, like "pinching" a hold with your heels, or literally jumping from hold to hold (doing a "dyno"). Style is important, but what really counts is to reach the last hold (the "top")
Bouldering was developed in the 60's in France and in California (by John Gill, the "human fly") as a pastime for bored rock climbers in off-season. In the 90's, however, the growth of indoor climbing walls (often homemade) and rock climbing gyms has made bouldering popular with the urban / hip-hop crowd, transforming it - in France and UK - in a "signature" sport like skateboarding. Part of the popularity of bouldering is due to the fact that it's being quite cheap - the only thing you need is a pair of climbing shoes (and the crashpad if you're bouldering outdoor).
There's a lively competition circuit around the world, and there are now bouldering sites (often simple concrete walls) almost in every city of the world. In fact, bouldering is now more a urban sport than an outdoor activity.
I'm going to a bouldering event tonight