Alpha Waves are electric neural oscillations in the human brain, that can be recorded by EEG. They indicate a lack of visual processing and a lack of focus.
In hypnosis, meditation, and other "altered states", these brain waves are seen as beneficial and desirable, despite evidence that they occur mostly under unpleasant or un-relaxed conditions, as an indicator of a loss of visual processing.
Alpha waves have also been used in handwaving proofs by medical woo and brainwave woo practitioners. There is no evidence to support that alpha waves assist in the regeneration of the body while asleep, or that they are in any way beneficial to thought. In fact, the lack of focus they indicate would seem to indicate the contrary.
There are also new age claims that inducing a Schumann Resonance at alpha wave frequency -- presumably by having everyone on the planet meditate at the same time -- will heal mother Earth and bring about peace and harmony and stuff.
You can, however, control a computer with your alpha waves.
- The Skeptic's Dictionary - alpha waves
- Controlling Computers with Neural Signals from Scientific American Magazine, Oct. 1996