(covers information from several alternate timelines)
A haircut, hairstyle, or coiffure was a specific fashioning of an individual's hair; the difference lay in whether it was trimmed into a particular shape (cut) or merely arranged differently (style).
One could have one's hair parted on the left, the right (more on one side than the other) or in the middle. In 2152, after apparently offending the Kreetassans for the second time, Jonathan Archer suggested that maybe his hair was parted on the wrong side. (ENT: "A Night in Sickbay")
Many races favored specific types of hairstyles or cuts, such as Vulcans, who often had a bowl cut they happened to share with Human 20th century comedian Moe Howard (ENT: "Carbon Creek"), and Romulans, who had a similar cut with pointed fringes overlapping their forehead. DeSeve wore his hair in the Romulan fashion following his defection to the Romulan Empire. (TNG: "Face of the Enemy")
Some hairstyles incorporated bangs, a section of short hair combed over one's forehead. In 2259, James T. Kirk vented about his frustration with Spock to Nyota Uhura, expressing his desire to "to rip the bangs off of his (the Vulcan's) head". (Star Trek Into Darkness)
One popular style was the beehive, favored by Starfleet women in the 23rd century. Jadzia Dax wore her hair in a beehive to blend in while time travelling to 2268 to help thwart "Barry Waddle"'s plot to kill James T. Kirk. (DS9: "Trials and Tribble-ations")
In 2366, Doctor Beverly Crusher reminded son Wesley Crusher that he was supposed to be getting a haircut, interrupting his conversation with Data in which he suggested Lal be sent to school with the other children on USS Enterprise-D. (TNG: "The Offspring")
While on the surface of the Ba'ku planet in 2375, Captain Jean-Luc Picard told Worf he needed a haircut. Worf experienced jak'tahla and told Picard that accelerated hair growth was normal during this time. (Star Trek: Insurrection)