Search engines use computer programs called Robots to automatically go from page to page through the web, reading content, and adding it to their databases. To speed up the process of getting your site indexed, they usually have a way for you to submit your site for indexing. You usually only have to tell a Search Engine the URL (address) of your site and it takes care of the rest.
Directories are run by humans who review web sites and categorize them within their directories. This leads to a more abridged set of sites which can good or bad depending on what you're searching for.