What is an authority site?
An authority site is a particularly high quality website or resource.
An authority site could technically be a commercial website that is the authority on a topic, though for many SEOs in certain contexts (ie for the purposes of writing an article) an authority site is generally considered to be either governmental, an association, or otherwise neutral from a commercial perspective.
Authority site examples
Examples of British authority sites would be Gov.uk, NHS.uk, BBC.co.uk, and others depending on the subject.