Which political system is run by a group of noble families or wealthy members of society?

An Oligarchy. An Oligarchy involves the political power being controlled and organized by a small group within a society. This small group may be families that own a large amount of wealth, military power, nobility, education, etc. in comparison to the majority of the population. Often, Oligarchies involve oppression to obtain obedience by the public. There are certain examples in which power has been passed down through families to keep the power from being shared to transferred from family to family in order to ensure political consistency. While the US has been considered a democracy, there are some who consider it an oligarchy, as a very small number of people who control a majority of the wealth have strong political and social influence in the nation. Aristocracy is very similar to Oligarchy, but the difference is that in Aristocracies the power is always passed down to new generations of family members.