A referring domain is a distinct website that links to yours. If one blog links to you from ten posts, that's ten backlinks but one referring domain.
Referring domains are usually a better measure of progress than total backlinks, because a new site linking to you is a new, independent vote.
Most backlink tools report both. When tracking outreach results, count new referring domains per month.
Example
Margin has 140 backlinks from 38 referring domains, because a few blogs link to it from many posts.
Putting it into practice
Where to use it: Link building cost calculator and Outreach volume planner. For the longer version, read finding and winning your competitors' backlinks.