Authority: Sometimes known as "domain authority," authority is a measure of a page or domain's rank in search engines. A high-ranking website with a large amount of authority will often be an older, more established site with a long history, useful backlinks, and unique, quality content on its pages.
A similar term, topical authority, refers to websites that are considered credible sources about a particular subject and are often cited on the internet. For instance, a popular brand name or well-known academic's website would both be considered topical authorities in their respective fields.
Wikipedia is a good example of a website that is considered a broad general authority site because it has useful information about so many different topics. A site about a specific topic would be considered a topical authority; for instance, ESPN would be a topical authority on sports.
As search engines develop their rankings, they will often adjust their formulas for determining a page's authority. For example, Google's Penguin algorithm seeks to weed out high-quality pages from those that are lower quality or use black hat SEO tactics, like link exchanges. Spam websites are a good example of pages that lack domain authority and are often banned from Google's results.
UPDATE TO THIS POST:On 1/19/2016, we switched from Moz to A h r e f s for domain metrics in the dashboard. Please see our Moz to A h r e f s system update post.