While many will use web directory and search engine interchangeably, the two terms mean something very different. A search engine is an automated piece of software that crawls through the internet indexing information.
A web directory, on the other hand, is something akin to the Yellow Pages of the internet. A listing of websites, under a category, put together by humans. A web directory will not be as complete when it comes to results, but it is a more curated listing of sites.
These are generally put into categories that go from broad to narrow, the more you hone in on what you are looking for.
If you want to get on to a web directory either submit your site to the owner of the web directory or hope they find your webpage and add it naturally.