Authority generally means being a recognized leader in your field or industry, which is useful when the goal is to rank well organically. Google's algorithms deploy machine learning to help make sure the content recommended to people is trustworthy, authoritative, and an expert is writing it.
High quality web pages should elicit meaningful interactions with users. Engagement can be quantified through engagement metrics such as time on page, bounce rate, pages per session, unique & new visitors, average session duration and so on. Regardless of whether or not a search engine deploys these metrics as apart of their algorithms, quality content and user experience play a massive role in SEO. Prioritizing audience centric content should have a positive effect on your engagement and conversion rates.
Trust & Reputation
Reputation can be more difficult to quantify, let's think about the people, sites, & institutions whose reputations you hold in high regard. They likely established that reputation over a long period of time by providing valuable & reliable information. So much so that they are cited by others as experts in their field and their colleagues or customers speak well of them.