A Board of Inquiry was a panel of ranking officer who judge serious but non-criminal offenses. An inquiry was usually headed by three officers of command rank, who conduct the questions and decide the outcome of the inquiry. They could acquit or sentence the defendant but could not deprive him of his liberty, including dishonorable discharge from Starfleet.
Any accusation made towards, a senior, commanding or flag officer is referred to a board of inquiry. If the boards judges that the charges are insignificant or lack evidence they can dismiss it without any witness heard. In the case a criminal offence is established, the case gets referred to a court martial. (Decipher module: Starfleet Operations Manual)
In 2268, Captain James T. Kirk convened a Board of Inquiry aboard the USS Enterprise to review the charges of misconduct against Ensign Pavel Chekov while he was infected by the Alnathian jewel. (TOS novel: The Klingon Gambit)
Superintendent of Starfleet Academy, Rear Admiral Brand, ordered a board of inquiry to look into the events surrounding the fatal accident involving the Nova Squadron in 2368. (TNG episode: "The First Duty")