Authority scores compress a site's backlink profile into one number, usually from 0 to 100. Ahrefs calls it Domain Rating, Moz calls it Domain Authority, and Semrush calls it Authority Score. Each tool calculates it differently, so the numbers aren't comparable across tools.
Google doesn't use any of these scores. They're useful as a quick filter when you're comparing prospects: a link from a site with a score of 70 usually carries more weight than one from a site with a score of 5, all else being equal.
Don't over-rely on them. A relevant post on a modest site that ranks for your topic can be worth more than a link from an unrelated high-score site, and scores can be inflated by spammy links.
Example
When choosing between two posts about note-taking apps, a site with a score of 45 that ranks on page one for the topic is usually a better target than a score-80 general news site that never covers software.
Putting it into practice
Where to use it: How to choose sites to pitch and Link prospecting search strings. For the longer version, read why authority scores are only part of link quality.