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Alpha waves are oscillating electrical voltages in the brain. They oscillate in the range of 7.5-13 cycles per second. Because alpha waves occur in relaxed states such as meditation and under hypnosis, they have been mistakenly identified as desirable. Alpha waves also occur under unpleasant conditions and when one is not relaxed. Research has linked alpa waves to throbbing pain. Alpha waves are not a measure of peace and serenity, nor are they indicative of an altered state of consciousness. Alpha waves are indicative of lack of visual processing and lack of focus: the less visual processing and the more unfocused, generally the stronger the alpha waves. If you close your eyes and don't do any deep thinking or concentrating on vivid imagery, your alpha waves will usually be quite strong.
There is no evidence that "When asleep, the brain goes into a repair and rebuild mode under alpha wave energy," as an ad for a protein supplement claims. Nor is there evidence that the brain is more insightful, creative or productive while producing alpha waves. Some think that increasing alpha waves can enhance the immune system and lead to self-healing or prevention of illness. This belief seems to be based on the notion that since alpha waves increase while meditating and relaxed, they are indicative of lack of stress, which can only be good for you. However, alpha waves can occur when one is not relaxed. Hence, increasing alpha waves is no guarantee that one is reducing stress, much less is it proof that one is enhancing one's immune system.
You can, however, learn to control a computer using your alpha and mu waves (the latter appear to be associated with the motor cortex because they diminish with movement or the intention to move). You can even compose music with your brain waves.
books and articles
Beyerstein, Barry L. Chapter 4. "Pseudoscience and the brain: Tuners and tonics for aspiring superhumans," of Mind-Myths: Exploring Everyday Mysteries of the Mind and Brain, Sergio Della Sala, M.D., Ph.D., ed. John Wiley and Sons, Ltd. 1999. (pp. 59-82).