Moving categories widget

Yes you can do this. You’ll first need to create a new widget area though.

Here’s more info on how you can do that.

http://codex.wordpress.org/Widgetizing_Themes

Then once that’s done you can then you’ll need to add the code to whichever template file you want to add it to.

You’ll also need to do a lot of css styling.

I think that Twenty Ten or Twenty Eleven has more than one widget area. Have a look and see how it’s done there.

There’s also quite a few tutorials on how to do that. Here’s one I found – http://www.wpreads.com/2013/03/creating-custom-widget-area-in-wordpress-theme.html