This page contains information regarding Star Trek: Short Treks, and thus may contain spoilers.
A board of inquiry was an official investigation by representatives of an agency, such as Starfleet. Such a Board would be empowered to investigate an event or person's activities for any alleged wrongdoing or criminal activity. After its investigation, it would issue a report and suggest possible action to prevent such wrongdoing from occurring in the future. It also would, if necessary, suggest punishment for any persons found to have committed or allowed such acts of wrongdoing to take place.
A board of inquiry investigated Captain Lynne Lucero for the loss of her starship, the USS Cabot, after only two weeks under her command and due to the actions of a single crewmember. That crewmember, Edward Larkin, had genetically engineered tribbles to increase their reproductive rate against Lucero's orders, which led to them exponentially reproducing out of control. The tribbles could not be contained and literally filled up the ship, forcing Lucero to order the crew to abandon ship. Larkin, who Lucero termed "an idiot", was the only casualty after he refused to abandon the Cabot before its destruction and was crushed by the reproducing tribbles he had created. (ST: "The Trouble with Edward")
Nilz Baris threatened to bring Captain James Kirk before a board of inquiry regarding his actions on Deep Space Station K-7. Baris claimed the Board would "roast him alive" due to his alleged improper actions in protecting a shipment of quadrotriticale bound for Sherman's Planet. (TOS: "The Trouble with Tribbles")
A board of inquiry in the mid-24th century investigated Captain Darson of the USS Adelphi for his actions after the Ghorusda Disaster. The board found the captain to have acted with "carelessness" during the disaster, which resulted in the deaths of 47 people, including Darson. It also found that Tam Elbrun, while not responsible for the deaths, should have been more aware and competent. (TNG: "Tin Man")
A board of inquiry was convened by the Superintendent of Starfleet Academy, Rear Admiral Brand, to investigate a fatal accident involving the Nova Squadron flight team in 2368. The board took statements from surviving flight team members Wesley Crusher, Sito Jaxa, Jean Hajar, and team leader Nicholas Locarno concerning the death of their teammate Joshua Albert. The board had been prepared to merely issue reprimands to the cadets, concluding that there was insufficient evidence to find that they lied in their statements. However, Wesley admitted that the team tried to execute an illegal Kolvoord Starburst maneuver, and further lied to the board to cover up the accident. (TNG: "The First Duty")
When Kashyk mentioned to Captain Kathryn Janeway that she'd violated the Prime Directive by rescuing a group of Brenari telepaths in 2375, she said, "Well, let's just say I usually go with my instincts and sort it out later at the board of inquiry." (VOY: "Counterpoint")