A daydream is a fantasy one has while awake.
After being bonded together as a result of mating and subsequently separating, T'Pol and Charles Tucker III began to daydream about each other. These daydreams were "accompanied by intense auditory and visual sensations." (ENT: "Affliction", "Bound")
In 2370, Captain Benjamin Sisko visited Major Kira Nerys while she was staying at Bareil Antos's monastery on Bajor, having been temporarily stripped of her post on Deep Space 9. Noticing he had disturbed her, he apologized for interrupting her meditation, but she stated she was simply daydreaming. Sisko commented bemusedly that he remembered doing that a long time ago and she replied that it comes back to you like riding a bicycle. (DS9: "The Circle")
The Talaxians believe that fantasies and dreams come from another place, slipping into the mind and whispering about things one has never imagined. They have a saying: "The dream dreams the dreamer."
In early 2376, The Doctor experimented with adding the ability to daydream into his program. However, the additions malfunctioned after Phlox, a member of the Hierarchy, covertly tapped into The Doctor's cognitive subroutines to spy on USS Voyager. This caused the daydreaming algorithms to intrude into The Doctor's perceptual subroutines, taking over his program. The Voyager crew was able to repair the damage, but later Phlox activated the daydreams again to warn The Doctor of an impending attack by his species. (VOY: "Tinker Tenor Doctor Spy")
The Doctor's fictional EMH counterpart in the holonovel Photons Be Free had 42 gigaquads of memory devoted to daydreams, something that disgusted Captain Jenkins when she discovered it, as she felt the activity was superfluous. (VOY: "Author, Author")