A web directory is a classification of websites under appropriate categories and sub categories. When a site gets submitted to a directory, it has to go through the process of review and approval by a human editor. The editor checks the website for quality, content, category selected and various other parameters. If the site matches these parameters, it gets listed in the directory under the category deemed to be most appropriate by the directory editor.
When a site has been listed in a directory, it makes it easier for the search engines to classify the site based on the key terms/phrases that have been used in the titles with which the site has been submitted. This gives your site a good chance of ranking high for searches targeting those terms. The higher your site ranks, the more visibility it gains, which consequently leads to an increase in the number of incoming visitors to your site.