East Slavic nobility worked different ways at different times, but with respect to The Idiot, what you need to know is basically this:

The Rurikid dynasty's succession worked roughly according to this thing called the rota system, dating from the reign of Jaroslav the Wise (d. 1054). Roughly, when the Grand Prince, who reigned in Kiev, died, his brothers would, in principle, all relocate themselves; the second-most senior prince would then move to Kiev, the third-most would replace him, and so on, and the oldest unlanded son whose father had lived to hold some land within this system would inherit the now-empty seat at the very end of the line. This is sort of an idealization, and it didn't last long: certain lands were in short order held out of the rota system and granted to certain branches of the Rurikid dynasty.

The title knjaz' 'prince' was basically usable by any male-line descendant of Rurik (i.e., any Rurikid), as well as descendants of Lithuanian Grand Dukes, and boyars too. This is where Myshkin's title comes from. He wouldn't be the only prince--his father, brothers, sons, and any other male-line relative would also be entitled to the style. As you can imagine, there were a great many people entitled to the title 'prince' by the 18th century.

So, the title knjaz' isn't really equivalent to English prince--a knjaz' was never really necessarily sovereign. At various times you would have a senior velikij knjaz' 'grand prince', in Moscow or Kiev, who was (nominally) in charge of many knjazi, but they weren't necessarily related to him. When Ivan IV crowned himself Tsar, the term 'grand prince' then became a generic title for members of the royal family. So, we use the English term 'prince', but the Russian term knjaz' is basically just a sort of general title for nobility.