Domain Authority is SEOmoz's calculated metric for how well a given domain is likely to rank in Google.com search results. It is based off of the Linkscape web index and includes link counts, mozRank, mozTrust, and dozens more. It uses a machine learning model to predicatively find an algorithm that best correlates with rankings across thousands of search results that we predict against.

It's best to use Page Authority (PA) and Domain Authority (DA) as comparative metrics when doing research in the search results and determining which sites/pages may have more powerful/important link profiles than another. While more specific metrics like mozRank can answer questions of raw link popularity and link counts can show the raw quantities of pages/sites linking, the authority numbers are high level metrics that attempt to answer the question "How strong are this page's/site's links in terms of helping them rank for queries in Google.com?"

Unlike other SEO metrics, Domain Authority is difficult to directly influence. This is because it is made up of an aggregate of metrics (mozRank, mozTrust, link profile, etc...) that each have an impact on this score. This was done intentionally because this metric is meant to approximate how competitive a given site is in Google.com. Since, Google.com takes into account a lot of factors, a metric that tries to calculate it must incorporate a lot of factors as well.

This means the best way to influence this metric is to improve your overall SEO. Particularly you should focus on your link profile (which influences mozRank and mozTrust) by getting more links from other well linked to pages.