Rapid eye movement, or REM, was the stage in the sleep cycle where dreams occurred in many humanoid species. Unlike REM sleep, delta sleep was a deep sleep cycle where dreaming does not occur.
Betazoids reached REM sleep at a different frequency from other humanoids. (TNG: "Night Terrors")
When a humanoid was unable to enter into REM sleep for a prolonged period of time, they might suffer from paranoia or hallucinations. (TNG: "Night Terrors")
While raising the mimetic simbiot Sim, Doctor Phlox explained in a letter to Dr. Lucas how he was at a loss to understand how Human adults could endure the REM cycle interruptions that accompany the teething stage of a Human infant. (ENT: "Similitude")
In 2367, the Tyken's Rift being tried to communicate with the crews of USS Brattain and the USS Enterprise-D during their sleep, and was responsible for the loss of the REM sleep, and eventual madness and death of the Brattain crew. (TNG: "Night Terrors") When Commander William T. Riker wasn't able to sleep properly, due to tampering by solanogen-based lifeforms, Doctor Crusher first assumed that his problems were caused by a lack of REM sleep. (TNG: "Schisms")
In 2377, Harry Kim asked Seven of Nine if the information she discovered about Unimatrix Zero was part of a dream. The Doctor disputed that, claiming she never reached REM sleep. (VOY: "Unimatrix Zero")