Short answers, pulled from the story.
A prince-bishop is a bishop who also serves as the civil ruler of a secular principality, distinct from the honorary Prince of the Church title held by cardinals. The office let a single figure govern land, taxes, and courts while also leading a diocese.
The Bishop of Urgell in Catalonia is the last prince-bishop, remaining ex officio one of two co-princes of Andorra. He shares that role with the French head of state, currently its president.
The title went defunct across the former Holy Roman Empire when the Empire dissolved in 1806 and its successor, the Confederation of the Rhine, was formed. A titular version persisted in Habsburg and formerly Polish lands until the pope abolished it entirely in 1951.
Ottonian kings, especially Emperor Otto I, granted loyal bishops Imperial lands and regalia privileges to weaken the power of hereditary dukes. Because bishops could not pass titles to children, this let emperors control territory without creating new hereditary rivals, a policy that fed into the Investiture Controversy of 1076.
The Prince-Bishops of Montenegro were the bishops of Cetinje, who replaced the region's secular Grand Voivodes in 1516 as Slavonic Orthodox rulers under Ottoman suzerainty. Their rule eventually secularized into hereditary princes from 1852 and then kings from 1910.
Three prince-archbishops held the rank of prince-elector: Mainz, Cologne, and Trier, confirmed by Emperor Charles IV's Golden Bull of 1356. Each of the three also carried the title Archchancellor over part of the Empire.