Sleep is a state of rest when the eyes are closed (for most species) and consciousness is altered. Dreaming takes place once a lifeform reaches REM sleep.
Unable to reach REM sleep within Tyken's Rift, the crew of the USS Brattain went insane and seemingly murdered each other. (TNG: "Night Terrors")
When he was briefly stripped of his powers and made a normal, mortal human, Q found sleep to be one of the most disturbing processes of the human condition. (TNG: "Deja Q")
Because he was an android, Data did not require sleep, a fact that disturbed Captain Picard when he was trying to fall asleep on a Klingon vessel headed for the Romulan Star Empire on a secret mission in 2368. (TNG: "Unification I")
In 2371, Bajoran First Minister Kalem Apren died in his sleep due to heart failure. (DS9: "Shakaar")
In 2376, an exhausted and sleep-deprived Tuvok fell asleep on the bridge of the Voyager, revealing that Vulcans, like Humans, snore. (VOY: "Muse")
Species 8472 do not sleep. (VOY: "In the Flesh")
Tosk only require 17 minutes of sleep per rotation. (DS9: "Captive Pursuit")