DMOZ was the Open Directory Project - a project whose aim was to catalog the web using human editors.
It was pretty difficult to get on DMOZ. You had to choose your category (a job in itself), provide a description, then wait until a DMOZ editor reviewed it.
If you got the OK, your website was added to the index. But often, after waiting months and months, your submission would be rejected. If you'd included a minor error, spelling mistake, sales copy, or if your site was 'too new', the editor would give it the thumbs down.
Getting on DMOZ gave you an instant boost on the search engines, because it was an indication of credibility.
Google would even sometimes use the DMOZ description in its results if it couldn't find something it liked on the web page.
I suppose it could never last, given the explosion in the number of websites and the bottleneck of human editors.
DMOZ closed on 17 March 2017.