Brainwaves occur all the time, not just at inspired moments. They are the electrical activity inside your head creating your thoughts and mental outlook. The dominant frequency at any particular moment determines your ability to perform a particular task. There are different optimal frequencies for different tasks, a summary of which is below. Learning to switch between these frequencies is, we believe, the key to peak performance.

Beta: (14-24 Hz) High level of mental activity often associated with decision-making, logic and problem solving.
SMR: (12-14htz) Quiet body, active mind. Ideal for attention, sequencing, information storage and retrieval. Ideal for optimal sports performance.
Alpha: (8-12htz) Relaxed but alert state of mind. Resting wakefulness.
Theta: (4-8 htz) Optimal state of mind for heightened creativity, sustained inspiration.
Delta: (Less than 4 htz) Deep physical and mental relaxation. 

The measurement of brainwaves began in the 1930s (Adrian 1934, Dietsch 1932) when a German scientist, Hans Berger, detected alpha rhythms in humans. More information on this subject can be found in the resources section.