A day was the amount of time it took for a planet to spin once on its own axis. This resulted in a day/night cycle (with "day" in the second case meaning the sunlit portion of the full day, also called daytime).
On Earth, a full day was divided up into twenty-four hours, whereas on Deep Space 9, a full day was divided into twenty-six hours. (DS9: "Afterimage")
The Andorians were at the Vulcan monastery at P'Jem twice, but left both times within a day because they were not able to prove their claim that the site was housing a secret listening post. (ENT: "The Andorian Incident")
When Sub-commander T'Pol was asked by a Vulcan elder how long she was aboard Enterprise NX-01 during their visit of the monastery at P'Jem, she told him she was aboard for nine weeks and four days. Later, she told the other members of the landing party that it would take days to explore the site of the monastery. (ENT: "The Andorian Incident")
Doctor Phlox estimated that the treatment of Captain Keene by using cortical regeneration would last two days, maybe three. Enterprise was sent to assist the ECS Fortunate as it was the nearest and fastest Starfleet ship, arriving at the scene in one day and a half at maximum warp. (ENT: "Fortunate Son")
On an uninhabited planet that Enterprise NX-01 explored in 2152, night only occurred on four days a month due to the planet's binary stars. (ENT: "The Catwalk")
In 2263 of the alternate reality, USS Enterprise Captain James T. Kirk told Doctor Leonard McCoy, shortly after Kirk returned from an unsuccessful away mission to the planet Teenax, that it was "just another day in the fleet." Kirk subsequently made a log entry in which he reported that the Enterprise was on its 966th day in deep space, and that telling where one day ended and the next one began was increasingly difficult. This was around a couple of days before his thirtieth birthday. (Star Trek Beyond)
The rotation period on J'Gal was 1.88005 days. (SNW: "Under the Cloak of War", display graphic)
Solar day[]
A solar day was a unit of time likely made up of solar hours.
In 2267, Admiral James Komack ordered the USS Enterprise to alter its flight plan accordingly to accommodate the advancement of the inauguration ceremonies on Altair VI by seven solar days. (TOS: "Amok Time")
Later that year, the estimated repair time to the damaged on the Enterprise's warp drive and deflectors was (at least) one solar day following its initial engagement with the planet killer. (TOS: "The Doomsday Machine")
After Norman had hijacked the Enterprise in 2268, he informed the crew that they would continue on their present course to the planet Mudd for approximately four solar days. (TOS: "I, Mudd")
After Spock had contracted choriocytosis in 2270, the nearest strobolin supply planet to the Enterprise's position at the time was Beta Canopus, which was located four solar days away at maximum warp. (TAS: "The Pirates of Orion")
Standard UFP solar days were mentioned in the Treaty of Armens. (TNG: "The Ensigns of Command", okudagram)
Parts of the day[]
A day was divided into time units including morning, midday, afternoon, evening, and night. Midnight was a specific time on a day. (ENT: "The Catwalk"; DS9: "The Nagus", "The Storyteller", "Defiant"; VOY: "Resistance")
The Novans also divided the day. Jamin asked Doctor Phlox how long of a day the treatment of Nadet would take and Phlox told him just a few hours. Later, Jamin threatened Captain Jonathan Archer that Malcolm Reed would be gutted if he and Nadet weren't back at the underside before day. (ENT: "Terra Nova")
Vice admiral Maxwell Forrest called Captain Jonathan Archer at four in the morning on one day in August 2151. Archer assumed it was not to talk about the data of Archer's Comet but to call Enterprise back to assist the ECS Fortunate. (ENT: "Fortunate Son")
The song "Heroes", which was used as incidental music in "Children of Mars", fantasized about the subject and their lover being heroes for just one day.