. In some cases, the monarch is elected. These exceptions make it difficult to define "monarchy" precisely; the most objective and comprehensive (albeit circular) definition would seem to be that a monarchy is a government that calls itself a monarchy.
Royalty is a Government in which the attention of the nation is concentrated on one person doing interesting actions. A Republic is a Government in which that attention is divided between many, who are all doing uninteresting things.
Americans also seem to believe that the monarchy is a kind of mediaeval hangover, encumbered by premodern notions of decorum; the reality is that the British monarchy, for good or ill, is a modern political institution — perhaps the first modern political institution.
A monarch's neck should always have a noose around it. It keeps him upright.
The monarchy is a political referee, not a political player, and there is a lot of sense in choosing the referee by a different principle from the players. It lessens the danger that the referee might try to start playing.