The most common typology of authority is that created by Max Weber. Weber argued that there are three kinds of authority. They are charismatic authority, traditional authority, and rational-legal authority.
Charismatic authority comes from the innate qualities of the person who holds that authority. The person does not hold authority because they were elected to some office. They do not have authority because they come from a given family. Instead, they hold authority because they have personal qualities that make other people want to obey them. Leaders of cults often have this sort of authority over their followers.
Traditional authority comes from unwritten rules that have been maintained over time. In this case, a person has authority because that is how things have always been. For example, the leader of a tribe of people may have authority because his family has been the dominant family for as long as anyone can remember. There is no constitution that says he has authority, but the unwritten rules of the society give him that authority.
Rational-legal authority is the kind that we have in democratic governments. This authority is based on written rules. President Obama, for example, has the authority to veto laws because there are written rules that specifically give him that authority.
This is the most commonly-used typology of authority.