An inquiry was a type of investigation into a formal complaint as to whether or not a crime was committed. The accused sits before a board of inquiry, who officially reviewed the evidence presented for and against the accused. If the case was not dismissed, and sufficient evidence was found against the accused, proceedings such as a court martial or trial would follow. These may be classified as, among other things, an informal or contrarily, official or formal inquiry.
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 took testimony from Captain Lynne Lucero for the loss of her starship, the USS Cabot, after only two weeks under her command. Protein specialist Edward Larkin had genetically engineered tribbles to increase their reproductive rate against Lucero's orders, which led to them exponentially reproducing out of control aboard the starship. The tribbles literally filled up the ship, forcing the crew to abandon the vessel. 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")
When, in 2266, Captain James T. Kirk was accused of committing willful perjury in his sworn deposition by Commodore Stone, following the apparent death of Benjamin Finney, deemed as culpable negligence. For this, Kirk was confined to Starbase 11, pending an official inquiry, which was set to determine whether a general court martial was in order for the aforementioned crimes. (TOS: "Court Martial")
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")