A private application is an application that is exclusively used by its creator (the owner) and not shared with other users. These applications typically implement user authentication, but may implement other types of authentication. The access tokens used in a private application may possess any privilege associated with the owner's ArcGIS account, and may even impersonate their ArcGIS account, allowing the app to act on their behalf and perform actions as them.
Private apps are not published or shared, and are often implemented as scripts that automate tasks related to the portal service. For example, a private script local to the developer's computer could perform tasks such as creating items, publishing data services, or administering an ArcGIS organization.