In more detail
Moz created Domain Authority (DA) as a way to compare the link strength of websites. Its explanation of the metric describes it as a comparative score calculated from link data, best used to compare sites with each other rather than as an absolute measure. Ahrefs offers a similar measure called Domain Rating, and other tools have their own.
These scores are logarithmic, so moving from 20 to 30 is far easier than moving from 70 to 80. They also differ between tools because each provider crawls the web separately and uses its own formula. A site can have a DA of 40 in one tool and a different number in another.
Most importantly, Google doesn't use any of them. Google's hiring guidance notes that third-party tools "don't have access to Google's internal ranking data", and it warns to be wary of tools claiming to be approved by Google Search (source).
Why it matters when hiring an agency
Authority scores are useful for rough comparisons: how your site stacks up against competitors, or whether a site that wants to link to you is reputable. They're a poor goal in themselves, because they can be inflated with links that don't help rankings or customers.
Be cautious if an agency's main promise is to "raise your DA" or sells links priced by the linking site's DA. That pattern is common in link selling, which Google treats as spam. Ask instead how authority work will support rankings for pages that matter to your business. For a page-level view, our Page Authority Checker scores individual pages and explains what affects the score.