An ‘Authority site‘ is a site that seems to get an extra ranking boost from the search engines. Somehow, in their analysis of the site and its backlinks profile, the search engine algorithm recognizes a site as being an ‘Authority’. Getting backlinks from sites that are seen by the search engines as an authority are close to being the sorts of links that probably past the most link juice, so they are much sought after.
Websites containing lots of informative content and delivering excellent user experience enjoy the status of authority sites.
To become an authority site, you have to match up the level of work done by such websites. For instance, wikipedia is considered as one authority site. It does not promote anything, it does not contain spammy content, neither it earns any money from the content published (in terms of ads or affiliates).