Dofollow and nofollow links: what the rel attribute does
A link is followed, or dofollow, when its HTML carries no rel value telling search engines to discount it. A followed link from an indexed page passes ranking signal to the page it points at. The rel attribute is how the linking site opts out of that.
The rel values and what each one says
rel="nofollow" asks search engines not to count the link for ranking. rel="sponsored" marks a link that was paid for. rel="ugc" marks user-generated content such as comments. Google treats all three as hints to discount the link. A link with none of these values is what people call dofollow; dofollow is not itself a value you can write.
What this changes for a directory listing
A directory that marks its product links nofollow still sends its own visitors to you, and its listings still appear in search when the pages are indexed. What the nofollow removes is the ranking benefit of the backlink. Whether that matters depends on why you wanted the listing; the point of publishing the attribute is that you know before submitting.
Directories change this setting, in both directions, without announcing it. When our crawler observes a directory's links flip, the change appears in the changelog.
How we verify it
We read the rel attributes on product links found on real listing pages, on a schedule. Each link we record is stored as one of four values: dofollow, nofollow, sponsored or ugc. What a directory says about its link policy is ignored; the markup we observed wins. The rules are on the methodology page.
A directory we have not read yet shows "unchecked" rather than a guess, and a directory can carry both kinds at once: the summary says what the majority of its observed links do, while the per-listing views keep each link's own attribute.
Where to see it: each directory's page in the index states whether its product links were observed dofollow or nofollow, and the backlinks view shows the attribute on each of your own listings.