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.
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")
An impostor of Captain Jean-Luc Picard doubted that William T. Riker had the evidence to support his attempted mutiny before a potential board of inquiry in 2366. (TNG: "Allegiance")
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")
A board of inquiry was to have been convened against Beverly Crusher after she disobeyed a direct order from Captain Picard not to perform an autopsy of a Ferengi. (TNG: "Suspicions")
In 2370, William Riker was going to appear before an inquiry at Starbase 247 regarding his involvement during the Pegasus incident. (TNG: "The Pegasus")
Later that year, Picard threatened to put Lieutenant Ro Laren before a board of inquiry for lying to him about details of her mission to infiltrate the Maquis. (TNG: "Preemptive Strike")
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")