Most CSM such as Mambo, postnuke etc have their own mini directory, however this is not really and excellent way to run a professional web directory. I suggest you look at a few specialist directory scripts.
There is an excellent free directory script called PHP link directory created by Dave DuVall I used that script to create UK Small Business Directory and it works perfectly.

I will strongly advice against using a CMS if you want to create very good directory as CMS are principaly designed to be CMS directory features in CMS are not as robost as using and dedicated directory script/software.

I hope this helps.

PHPLinks is a good script, get yourself a script, then sit down and work on the taxonomy of it before adding any sites, as it is as hard as hell to re-structure later.

Oh no! Not another web directory. There are millions of them already

Anyway, as OldWelshGuy, phplinks is a good script to use. It is free and is among the most popular scripts used for web directories.
You can check out hotscripts.com to find other scripts.

Best of luck. And let me know when you set up your web directory so that I can add my links

Sorry but it would be only for small local business in my country.

Hopefully, it would not be "just another web directory".