DMOZ is another name for the Open Directory Project ODP. The name is derived from it's original domain name 'directory.mozilla.org'.
DMOZ is the largest web based, open content, multi-lingual directory of websites on the internet. It is human edited and is owned by AOL. The directory is used to (help) categorise and organize the information on the internet. A site can be submitted to DMOZ by visiting http://www.dmoz.org/ and submitting the site URL under the appropriate category. There may however be a significant delay of the order of months before the site is indexed (included). This depends on how active the category is (and how conscientious or energetic the category editors are).
While the DMOZ directory is reported to be used by Google the AOL search drives DMOZ (presumably due to AOL ownership of the directory). The DMOZ directory continues to grow as sites are submitted and (eventually) added to the directory. However, there is reported to be a significant drop in the number of visits to the site over time and the search frequency on DMOZ is surprisingly low.
You can find information about DMOZ traffic, statistics, top queries etc. on Alexa from http://www.alexa.com/siteinfo/dmoz.org.
How important is DMOZ to SEO?
As mentioned above, Google uses DMOZ (for it's back up directory source). As long as Google uses DMOZ and as long as Google is the market leader for search, it is vitally important to ensure that your site is submitted to DMOZ and to follow up on the listing until this is added.