Authority in digital marketing is the perceived power, trust and dominance of a website against the competition.
Search engines take in a lot of different metrics to determine a website's authority including age, inbound links, popularity and trust. Authority is a major factor in determining ranking and it requires a lot of good work to achieve a high authority score. It could be said that every activity an SEO undertakes has the ultimate goal of improving the sites authority.
Moz is credited for coining the term 'domain authority' to measuring the ability of one site to rank against another. A key part of Moz's domain authority scoring metrics are the inbound links coming from other authority sites.
Authority exists at both domain and page levels:
High authority domains effectively pass on authority when they link externally. It's in your websites interest to get as many relevant high authority links as possible. On the other hand, links from sites with low authority or from sites with no relevance to your website's content could damage your authority. You should monitor your links using a tool such as Majestic to ensure your inbound links stay in good shape.
Different pages on your site will have differing levels of authority depending on their links, content and popularity. It's good SEO practice to share this authority by linking internally from higher-authority pages to other related pages which might not be so authoritative.