I need to convert a WordPress cateogory ID to it's name for echo'ing on screen and get the URL for that category.

Users will have the ability to enter a category ID into their WP admin panel, then the theme will grab that ID, post the latest posts for that category and add a title and link to that category. I can do everything but work out the name and URL.

Any ideas on how to do this?

I had a good hunt through the WordPress codex but haven't had any luck figuring this out.