I am attempting to discover how you can style just the category widget. All icons are utilizing the same div names. It is possible to code I'm able to use to distinguish the course widget?

Your theme should enclose icons in certain tag having a class unique to that particular widget. Consider the documentation for register_sidebar based on how to specify the before_widget property to do this:

http://codex.wordpress.org/Function_Reference/register_sidebar

Failing that, when the widget does not have unique selectors one can market to, then you will need to create them. If you are likely to edit a built-in widget, I suggest copying its code to your functions.php instead of editing it in position, to ensure that improving WordPress does not remove your modifications. Just register the widget having a different title.