If you have posts that only should be displayed on category extract-type pages, then set your site to show a static page on the home page, rather than the latest posts.
This setting may be in your theme options, or in the Settings, Reading screen. (Some themes may override the Settings, Reading settings, so check both place.)
Added 2/28
If you want to have a page that only shows posts of a certain category, then creating a template for that category might be the answer. One way to do that is discussed here: http://www.wpbeginner.com/wp-themes/how-to-create-category-templates-in-wordpress/ . You can use that as a starting point.
There are also ‘category page’ plugins, but a template might be best.