Whitewater rivers are those broken with rapids, where the stream encounters steep gradients, boulders and other jams. Big rapids are typically found in the upper courses of rivers, where they are actively eroding highland terrain, but whitewater can be encountered on lower portions as well.
People have been traversing dangerous rivers on rafts for a long time. John Wesley Powell, for example, famously explored the Colorado River in 1869. Modern recreational rafting began to gain popularity when surplus inflatable life rafts came on the market after World War II.
By convention, rapids are rated by an international difficulty scale, ranging from modest, mostly safe riffles (Class I) to the extremely dangerous, near-unrunnable sections of Class VI rivers. A given river might shift classes depending on season and environmental conditions.
As the sport has matured, rafting options have expanded. Neophytes can sign up for highly secure guided trips, while experienced, adventurous boaters might tackle fearsome, never-run rivers in far-flung places. Whitewater trips can last an hour or many days.
People interested in rafting should familiarize themselves with the real danger that rivers present. If you're just starting out, hook up with a guided trip and stick to lower-class rapids until you're more experienced.