Each post in WordPress is filed under a category. Thoughtful categorization allows posts to be grouped with others of similar content and aids in the navigation of a site.
A slug is a few words that describe a post or a page. Slugs are usually a URL friendly version of the post title (which has been automatically generated by WordPress), but a slug can be anything you like. Slugs are meant to be used with permalinks as they help describe what the content at the URL is.
How do you link a category to the homepage or any section?
You can't "link a catgeory" to anything. However, WP does include a template tag that allows you to list links to all of your categories.
If your theme does not display links to your categories as soon as they contain at least 1 post each, then I think you should try looking for another theme. Editing your current theme - whilst possible - would involve some degree of coding and/or template editing.