Can You Identify The Three Main Types Of Authority In An Organization?

Your question is a little confusing, since you have mentioned types of authority and types of organization as well. The types of authority are different from types of organization. However here is a brief overview:
Types of Authority depends on level of management and is divided in three layers i.e
Top level
Middle Level
Functional Level
Whereas types of organization depend on their organizational structure and are as follows:
-Functional organization
- Matrix organization
- Divisional organization
For details see the link below
Organization

There are 2 main types of authority in an organization.
-Line Authority
        the position authority (given and defined by the organization )that entitles a manager to direct the work of operative employees

-Staff Authority
         Positions that have some authority (e.g..,organization policy enforcement) but that are created to support assist,and advise the holders of line authority.