A directory is a site where human editors can manually group websites into categories and give them descriptions. Websites are added by either the website owner or an administrator for the directory. This in different from, for example, a search engine directory where the information is collated by machines.
An example of a directory website is the Yelp business directory. Business owners can add there business listing and add a description, address, images, opening hours and other relevant information. Accurate business descriptions and good keyword choice in the profile descriptions is a quick and easy technique for business owners to optimise their site for local search.
Directories usually link back to the business website giving the search engines relevancy signals about the site. However, listing your local business on a low quality website, or providing conflicting or inaccurate information across a few directory listings, could give a negative signal or even get your website penalised.
Here's a handy list of local UK directory websites that you can get quality backlinks from. Remember to make sure your details are exactly the same across all listings so as not to confuse the search engines.