Short answers, pulled from the story.
A court-martial is a military court, or a trial conducted in such a court, empowered to determine the guilt of members of the armed forces subject to military law and to impose punishment. Courts-martial can also try prisoners of war for war crimes and, in some circumstances, civilian defendants, particularly in cases involving violations of martial law.
Courts-martial have jurisdiction over both civilian-type offences such as fraud, theft, and perjury, and purely military crimes such as cowardice, desertion, and insubordination. In the United States, offences are governed by the Uniform Code of Military Justice; in the United Kingdom and for Canadian forces, they are defined in the Armed Forces Act 2006 and the National Defence Act respectively.
Most navies convene a court-martial whenever a ship is lost not because the captain is presumed guilty of wrongdoing, but to place the circumstances surrounding the loss into the official record. The proceeding is documentary rather than punitive by default.
Harold Pringle was the last Canadian soldier executed following a court-martial verdict, put to death in 1945 after being convicted of murder. Capital punishment for military offences in Canada was abolished in 1998.
The format varies considerably. France has no courts-martial in peacetime and uses civilian courts instead. Indonesia organises military justice across four tiers matched to officer rank. Finland reserves formal courts-martial for wartime, handling military cases through specially composed civilian courts in peacetime. The United Kingdom made the Court Martial a permanent standing court through the Armed Forces Act 2006.
Herman Melville's novella Billy Budd, first published in 1924, centers on a drumhead court-martial for striking a superior officer, later adapted into Benjamin Britten's 1951 opera. C. S. Forester's 1938 novel Flying Colours features Captain Hornblower tried for the loss of HMS Sutherland. The 1992 film A Few Good Men focuses almost entirely on the court-martial of two enlisted Marines.