Directory or web directory is a website with a categorized list of websites that are created by human editors. Editors usually categorize websites by subjects, manually put in titles and relevant descriptions.
The term is often used together with search engines. However, website lists created by search engines are put together by technology while the directory is put together by real people. Search engines do not have bias and make lists based on relevancy and the number of views. Directories give a chance for small websites to be seen and evaluate relevance by actually reading the content rather than analyzing keywords usage and page views.
Some directories work on a free basis and sort out the content themselves. They still receive the applications from website owners but they have the right to say no. Paid directories charge for the link being included in the directory.
When link building was still a relevant option, many SEO professionals paid directories to get their link featured. It resulted in bed-moderated spammy directories that made link building irrelevant.
Yet, to this day you can gain more views through directories without hurting your SEO. If you use well-moderated directories like Best of the Web and your website is actually relevant for the subject, it will work. Google itself often brings directories to the top of the search results.