Short answers, pulled from the story.
The Crown is a political concept used in Commonwealth realms to denote the state and the exercise of governmental authority, including executive, legislative, and judicial functions. Canadian academic Philippe Lagasse described it as acting in multiple capacities: crown-in-council (executive), crown-in-parliament (legislative), and crown-in-court (judicial). At its most basic, it is what most other countries call "the state."
The Crown concept developed under the feudal system in England, evolving over several centuries as parliaments, ministries, and courts grew through the 13th century. It was first formally defined as an "imperial" crown during the reign of Henry VIII in the Ecclesiastical Appeals Act 1532.
The Crown was historically considered indivisible, but as the self-governing dominions gained control over the royal prerogative in the 1930s, it evolved such that the Crown in right of each realm acts independently. The Balfour Declaration of 1926 recognised the dominions as autonomous communities equal in status, and the Statute of Westminster 1931 enshrined that recognition in law.
The Irish Free State implemented the abdication of Edward VIII a day later than the United Kingdom and the other dominions, creating a 24-hour period during which Edward VIII was king in the Irish Free State while George VI was king everywhere else. This divergence exposed the limits of the Crown's legal indivisibility across separate realms.
In criminal proceedings, the Crown is the prosecuting party, with cases designated as R v the defendant's name, where R stands for rex (male monarch) or regina (female monarch). In Scotland, criminal prosecutions use the abbreviation HMA, standing for His or Her Majesty's Advocate, in place of rex or regina.
The theory of the King's two bodies holds that the sovereign simultaneously possesses a body natural, which is subject to infirmity and death, and a body politic, which never dies. This theory underlies the immediate succession of a new British monarch upon the death of a predecessor; the body politic of the Crown continues unbroken even as the individual monarch dies.