A link directory is a collection of web links arranged categorically. It can be arranged alphabetically too. If its a business site, the category will be business, if it's a travel website, the category will be travel or tourism etc. You have to submit your website to the link directory through the submit page. Often link directory websites charge fees for reviewing your website. Though many of them are free. But most of the free link directories are full of junk or spam websites. So always check the directory before submitting your website whether its a junkyard. Search engines specially google doesn't like bad neighbourhood.
Now if you are searching for a good free stable link directory script then I will suggest PHP Link Directory script. Though there are few more scripts available like esyndicate, indexu, golinks, cplinks, skalinks, PHPLD is always my favorite. It has both free and paid version. The main features of php link directory script are easy to install, search engine friendly urls, easy to modify templates, lots of free and customized templates, some nice free apps or widgets all of them made the script unparalleled. If you want to install the script by someone else, you can contact me. I do installation jobs for just $10 per installation.
There are thousands of free and paid link directories running in the world of web. So if you want to see your directory stands high, you have to work hard. Give at least 30 mins everyday for your php link directory. Make it an useful resource for visitors. Take your time, don't hurry. You can make a quality directory if you want, its not hard nor complicated. Moreover my best wishes are always with you.