A referring domain is a website that links to yours at least once. If a local newspaper links to your site from ten different articles, that is ten backlinks but one referring domain. Search engines care more about the number of different sites that vouch for you than about the number of links, so this is the count to watch.
The second link from a site you already have is worth far less than the first link from a new one. A report that boasts of a hundred new backlinks when the referring domain count moved by two is describing one thing happening a hundred times.
