So the British Monarchy (and all the monarchies of the Commonwealth) have what is called Primogeniture; the eldest living child of the ruler at the time of the ruler's death will inherit. (It goes further with a recent parliamentary declaration that gender of the eldest child doesn't matter). So, when Lizzy 2 dies, Chuckles gets the throne, unless he's dead, too. In which case it passes to the next kid. And so on and so forth.

In Saudi Arabia, the Crown Prince (heir apparent) is selected by the ruler of the House of Saud, but generally follows an issue of Seniority; it goes to the eldest member of the Saudi royal family, be that some uncle or brother or cousin or what have you of the recently deceased monarch. (It goes further to stipulate only men can inherit, and the inheritance can only be traced through the male line of the family.)