Images from Jean-Luc Picard's 2373 Borg-induced dream
A dream is a series of sensory inputs and emotions involuntarily occurring in the mind during REM sleep. Generally, only those sensory inputs that are pleasant or neutral to the dreamer are referred to as dreams; those that are disturbing or frightening are specifically known as "nightmares".
AIs sometimes had the ability to dream added to their programming. Noonian Soong added the ability to dream to Data. (TNG: "Birthright, Part I", "Phantasms")
Someone dreaming would often see people they knew act very different from their normal ways. In November 2151, aboard a damaged shuttlepod, with little hope of rescue (believing the Enterprise NX-01 to have been destroyed), Malcolm Reed dreamt that, while recovering in Enterprise's sickbay, he received amorous attention from T'Pol, impressed with his "heroic" actions, but was awakened just as the Vulcan was about to kiss him. Not long after, having actually been rescued, he asked T'Pol if she was supposed to "say something" about "heroics" - upon hearing her less-than-amused response, he smiled, as it signaled that he was not dreaming and had in fact survived. (ENT: "Shuttlepod One")
In 2367, the crew of the USS Enterprise-D went slowly insane as they were unable to dream while trapped in a Tyken's Rift. (TNG: "Night Terrors")
Worf once commented that Klingons dream about "things that would send cold chills down your spine and wake you in the middle of the night." (DS9: "Rejoined")
In 2373, Captain Jean-Luc Picard experienced a terrifying dream about his experiences with the Borg. The dream was apparently induced by the Borg's proximity. (Star Trek: First Contact)
In 2374, the USS Voyager encountered the dream species, a species of aliens that lived out their lives in dreams. They became trapped in the same shared dreaming state the aliens lived in, experiencing "false awakenings" - a phenomenon where one dreams they have awakened from a dream, only to have transitioned into another dream - while trying to wake themselves up and escape. (VOY: "Waking Moments")
In 2375, Nog asked Vic Fontaine if he dreamed when his program was off. In response, Vic only smiled ambiguously. (DS9: "It's Only a Paper Moon")
In 2376, The Doctor added a daydreaming subroutine to his program. (VOY: "Tinker Tenor Doctor Spy")
Background
Although stated as such in TNG: "Night Terrors", REM sleep is not the exclusive purview of dreams - it is simply the most common time to dream. 5-10% of people who are woken during non-REM sleep also report dreaming. We can perhaps forgive Crusher being inexact as she was impaired by the effects of the Tyken's Rift.