How to show certain categories on certain pages

Hi all!

I’ve been using WordPress for a while now and I have to followin problem. I have a menu that consist from pages. What I would like to do is that it would show different categories on different pages on the sidebar. And furthermore, when you click on the category link or the post, it would keep the same category links on the sidebar. And it should be possible to add these categories or change their name without touching the actual code.

If someone knows the solution, I would highly appreciate it! ??