Domain Authority (DA) is a metric used to evaluate sites based on how well they are likely to rank in search engines. It has become a replacement for Google PageRank, which was widely used in the early 2000s for this purpose, but is no longer published.
Domain Authority is was developed by Moz, and the best way to find out your site's DA is to head over to Moz.com's Link Explorer tool: https://moz.com/link-explorer. You'll have to sign up for a free Moz account to use this service.
The tool will analyse the site that you entered and provide a numeric score. You can also use the tool to compare the results for different sites.
If you select "exact page" rather than "root domain" the tool will give you a couple of numbers: Domain Authority, which measures the rank of the site as a whole, and Page Authority (PA), which measures the rank of a particular page -- this will probably be your homepage in your first test. You can also measure the PA of particular posts or pages by pasting the URL for that page in the search bar.
Scores range from 1 to 100, but since this tool is comparative, there's no "good" or "bad" score. Very well-regarded sites like Wikipedia or the New York Times will have high scores, and brand-new sites always start with a score of 1.