A formal reprimand was a disciplinary action taken by Starfleet against officers or cadets who undertook unauthorized actions. It usually took the form of a written statement placed in the offending officer's permanent file by their superior, noting the improper action for future reference. While a reprimand typically carried no other immediate punishment, it could affect the offender's future career opportunities, such as promotions to a higher rank or position.
In 2151, Hoshi Sato had a leave of absence from Starfleet while she taught in Brazil. When Jonathan Archer wanted to order her to return, she told him he could only forcibly recall her after a reprimand which in turn would exclude her from serving on an active vessel. (ENT: "Broken Bow")
In 2152, after losing his communicator on a pre-warp planet, Lieutenant Malcolm Reed told his commanding officer Jonathan Archer that he was prepared for any reprimand he would give him due to his carelessness. (ENT: "The Communicator")
Later that year, Travis Mayweather worried that he might be the recipient of a reprimand on his record after he became late for his duty shift on Enterprise NX-01 when Dr. Phlox kept him in sickbay for quite a long time obsessing over his headache. (ENT: "Singularity")
In 2368, Rear Admiral Brand put a formal reprimand in each of Cadets Wesley Crusher, Jean Hajar, Sito Jaxa, and Nicholas Locarno's permanent records after they covered up evidence that they participated in an illegal maneuver called the "Kolvoord Starburst", which cost the life of Cadet Joshua Albert. (TNG: "The First Duty")
In 2370, Lieutenant Commander Geordi La Forge was given a reprimand by Captain Picard for violating orders and reconnecting himself to the interface probe in an attempt to find his mother. (TNG: "Interface")
In 2373, Captain Benjamin Sisko stated that if Dulmur and Lucsly from Temporal Investigations wanted to put a letter of reprimand in his file for going up to James T. Kirk and speaking to him briefly in 2268, they were free to do so. However, Dulmur admitted that he probably would have done the same thing if given the opportunity. (DS9: "Trials and Tribble-ations")
In 2374, Lieutenant Commander Worf received another formal reprimand from Starfleet Command, after he failed to meet with Dominion informant Lasaran and instead took his injured wife, Lieutenant Commander Jadzia Dax back to Deep Space 9 for treatment. Captain Benjamin Sisko told him that, as Lasaran had information that could have saved millions of lives, Worf would probably never receive the rank of Captain. (DS9: "Change of Heart")