Rob Woods: A made up number that the search engines don`t use. It`s an attempt to approximate the power of all the links to a domain. Domains don`t actually have link authority in the search engines` eyes however, pages do.
Trey Collier: It is a made up metric (by SEO MOZ) that estimates the "Value" of a Domain as far as possible ranking power......and then +Rob Woods posted before I was finished. What he said.
Rob Woods: also, a way that dodgy link builders use to charge for links. The link authority of the page your link is on is all that matters as far as "authority". A link from a page with 0 authority on a DA 99 site, has 0 authority.
Roger Montti: There is no such thing as domain authority in use at Google.
Mark Traphagen: Several have already commented on how this has no value for links, and I agree. We use it for one thing and one thing only. Internally we use it as part of our metric for determining our "major mentions" online each month. We count only mentions of our brand on sites with a DA 50 or higher. Totally arbitrary, I know, but we needed a simple cutoff for determining which of many mentions might be significant.