REM, is one of the five stages of sleep that most people experience nightly. Its main characteristics are very fast and random movements of the eyes (hence the name) and also a very low muscular activity. REM is considered to be a very important sleep stage since it helps memory consolidation and provokes neural stimulation.
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You can find a lot more information in the related Wikipedia entry http://en.wikipedia.org/wiki/Rapid_eye_movement_sleep