Link Category Table

I need a little help and I don’t know if I’ll explain it right. I want to make a page for my links (I want to exchange links).
I want to make a page with a table on it. I want to be able to put categories, click the category and you will see the links in that category.
I’ve installed and uninstalled so many plugins for this, none of them were what I was looking for. Can anyone point me in a direction as to where to look?
Thanks in advance.
PS- On my site I have a link page but I can’t add my own links to it. And most links to exchange are already formatted the way they want it.
http://www.WeeklyDebater.com