(covers information from several alternate timelines)
Hair is a series of filaments of dead cells that grow on the skin of animals, usually mammals. In humanoids hair often grows on the top of the head, where it is styled for cultural reasons. On mammals, it was generally termed "fur".
In 2369 Jadzia Dax told Kira Nerys about Morn who was asking her for dinner. She declined but though that he was cute because of his seven or eight wiry hairs sticking out of his forehead. (DS9: "Progress")
When Lwaxana Troi asked Constable Odo about his hair, he told her that this was no real hair but a part of himself, the shapeshifter. She then asked curiously if he studied hairstyles, to which he replied that he imitated the style of the Bajoran who studied him. (DS9: "The Forsaken")
Notably without hair, the Bolian race have sometimes been seen to wear a toupée to hide this fact. It is unknown if this is a fashion trend, or simply a way to blend in with most of the other humanoid species while away from their homeworld, Bolarus IX. (DS9: "The Muse") Bolian barbers included Mot and V'Sal. (TNG: "The Host", "Data's Day", "Ensign Ro")
When Seven of Nine was freed from the Borg Collective in 2374, The Doctor was able to stimulate her hair follicles to re-grow. (VOY: "The Gift") This was done at least once more, when, in 2376, Lyndsay Ballard returned to the crew of the USS Voyager as a Kobali. (VOY: "Ashes to Ashes")