Neurofeedback is based on the principle that certain kinds of activity in certain parts of the brain correspond with particular kinds of problems, such as ADHD, depression, epilepsy, etc. By measuring and providing feedback on this brain activity, you can learn to control it voluntarily to some extent and reduce the symptoms of the problem you're having. It's still a very young procedure, but the research is promising. (In one study, for example, epileptics were able to reduce their number of seizures by a third.)

Here's an example of how it might work. Let's say we've got a kid with ADHD. His doctor is worried about his reaction to standard drugs, so he refers the kid for neurofeedback. The practitioner--usually a mental health professional of some kind--places small electrodes on the kid's scalp and those electrodes measure brain activity and feed it into a computer. On the computer screen is a picture of a helicopter. When the kid's brain waves are "hyper", the copter climbs, and when they're "calm" the copter drops. Eventually the child learns to control the brain waves to the extent that he's able to guide the plane through various obstacles, etc. (The kind of feedback you get varies...it might be a computer game, or a sound that you hear, or something completely different.)

The point is that the client learns to voluntarily control his or her brain activity to some extent. Hope that helps a bit.