Verizon owns the DMOZ.org domain name, but they shut down the DMOZ web directory in March of 2017.
At one time this page you are now reading contained detailed instructions on how to get your website listed in DMOZ, also known as The Open Directory Project, also known as the "ODP", which was (as the name suggests) a huge, volunteer-based open directory. Anyone could volunteer to be the 'editor' of a particular niche category within the directory, and have near total control of what showed up in that category after that.
Disclaimer: For several years I was a volunteer ODP editor in charge of an obscure literary category. The decisions of the editors were subject to review, of course, but like-minded supervisors could and did consign websites to wrong categories (on purpose, so they wouldn't be found), and would sometimes refuse to put websites that belonged in a particular category in that category, based solely on personal dislikes and prejudice. I observed this happen time and again with biased ODP editors having some stake in a particular niche, rejecting competitor websites that belonged in their fiefdom. If they assigned a website into a wrong category, it couldn't be found when someone was looking for it in the correct category.
Also, it could take months or years to get a website listed in a correct category, because ODP editors simply didn't perform their volunteer work with adequate speed.
Because ODP was a 'human-edited' directory, Google used to throw higher rankings toward sites listed within the ODP categories, Sort of in the same way Google now rewards WikiPedia articles (despite the fact that they are sometimes notoriously biased). Human nature being what it is, it's impossible to keep the articles from being slanted one way or another. Google's algorithmic bias places WikiPedia articles toward the top of their search results, even though many articles obviously do not appear to warrant (if you actually look at their content) such high placement in Google's search results.
In any event, the DMOZ/ODP directory is now ancient history. You can still see a snapshot of DMOZ as it existed just before it was closed down, and get a more specific history here of the life and times of DMOZ.
No, you cannot advertise in DMOZ now, and no, you cannot add your site to it now or change an existing listing in it now. It just is what it is, and will continue as a small chapter of internet history....