Something like a duke, and the title wasn't all that special.
The English word "prince" is translated from the Russian "knyaz (князь)", which could be used either to denote a member of the royal family or more commonly a member of the nobility. Men directly related to the Tsar were usually called Velikiy Knyaz or Grand Prince instead. "Knyaz" can also be translated as "duke", which would probably give a more accurate idea of Prince Myshkin's status.
As the Tsardom of Russia gained dominion over much of former Kievan Rus', Velikii Kniaz (Great Kniaz) Ivan IV of Russia in 1547 was crowned as Tsar. From the mid-18th century onwards, the title Velikii Kniaz was revived to refer to (male-line) sons and grandsons of Russian Emperors. [...]
Kniaz [...] continued as a hereditary title of Russian nobility patrilineally descended from Rurik (e.g., Belozersky, Belosselsky-Belozersky, Repnin, Gorchakov) or Gediminas (e.g., Galitzine, Troubetzkoy). Members of Rurikid or Gedyminid families were called princes when they ruled tiny quasi-sovereign medieval principalities. After their demesnes were absorbed by Muscovy, they settled at the Moscow court and were authorised to continue with their princely titles.
From the 18th century onwards, the title was occasionally granted by the Tsar, for the first time by Peter the Great to his associate Alexander Menshikov, and then by Catherine the Great to her lover Grigory Potemkin. After 1801, with the incorporation of Georgia into the Russian Empire, various titles of numerous local nobles were controversially rendered in Russian as "kniazes". Similarly, many petty Tatar nobles asserted their right to style themselves "kniazes" because they descended from Genghis Khan
I haven't found a good figure for how many knyazi there were in Russia around the time when The Idiot was set, but given the size of Russia, the variety of places a man could pick up this title, and the generally large families of the Russian aristocracy of the time, I would expect at least thousands. This is supported by a Russian Wikipedia page listing hundreds of Knyaz families.
Our hero Prince Myshkin isn't a royal figure, or one of the highest nobles in the country. He's more like a son of a decaying aristocratic family which has lost most of its money. This is also quite apparent from the text of the novel, IMO - the word "prince" is thrown around quite casually, and he's never treated with special reverence due to his title.
