Domain authority or DA is a score provided by Moz. More Domain authority you have more chance to rank on Google or any search engine.Search engine appears their search based on DA (It is not the only way to rank).The score is between 1-100. Moz calculated it by evaluating your website back-links. That means if you have more external links than your website domain authority will increase as per the quality of the links.
To search a website domain authority you can visit opensiteexplorer.org. But you can calculate only 3 times in a Day.You can Use Our Tool section to check your website DA.
You can Take Look at this picture, Domain authority of Amazon.com is 98.
A good domain authority score 20+. Websites like Facebook.com.Google.com. Linkdlin.com they have DA 100. Because they have millions of external backlinks from millions of websites.
So, To increase your Domain authority and rank on the Seach engine you need to build external links from different websites.