"Principalities are either hereditary,...or they are new." Hereditary principalities are those where rule has been established for generations, usually by one family. The people tend to love a ruler of this kind, and there's usually little opposition to his rule. Mixed principalities are those where the rule of a prince is new. He has acquired the new territory either through annexation, the wishes of the people, or the actions of the nobles.The problems a ruler encounters can be difficult. If he has taken the area by force, the people of the newly acquired area are going to fight the ruler at every opportunity. A prince has three choices in this case. He can totally destroy the territory, live in the territory among the people, or create a government in the territory that will support him, allowing the people a sense of self-government. A civil principality is one where the ruler is given power by the general population or by nobles. In this case, a ruler needs to gain the favor of at least a few of the nobles and most of the people to be successful. An ecclesiastical principality is one where the real power is held by religious people. It doesn't matter what the prince does because the people's belief in their religion is so strong.

According to Machiavelli in The Prince, what sorts of principalities are there?  

There are two kinds of principalities according to Machiavelli. They are hereditary principalities and new principalities. To define hereditary principalities didn't occur to Machiavelli, so there is some dispute over rules to be used to determine what constitutes an hereditary principality. On the other hand, it is universally agreed that a family that has never before exercise governing rule is a new principality. In light of this rather obvious definition of new rule, Machiavelli illustrates his point with the example of Francesco Sforza, the new ruler of Milan. The Dukedom of Milan was not recognized as resting with the Sforzas until Francesco's son, Ludivico, finally took the throne, at which time the Sforzas were recognized by the Holy Roman Emperor and by then by other principalities as well.