What is the importance of REM (rapid-eye-movement) in sleep?
Rapid Eye movement is a stage within the rest period (our sleep) in which our eyes reportedly move in a way that is described as "saccadic". The result of the saccadic movement of the eyes is that the brain neurons supposedly act at the same rate of activity during this time of the sleep process as they do when we are awake.
Another result of the REM sleep stage is that, according to scientific research, most of the subjects who were awoken in this stage as part of their experimentation were able to recall dreams. This is because of the level of activity of the neurons, whose main duty is to transmit the impulses from one cell to another and produce what we further extrapolate as information.
Check out the link provided in the answer so that you can get more acquainted with the REM process.