Possible to assign fields (or an image) to a category?
The Craft Categories page is very useful.

To assign an entry to a category, create a category field, and require it in the entry type (you are probably already doing this).

When you create a Category Group you can edit the field layout, and add an image field and a text field for the category. Then to edit that content, double click on the category itself (from the right-side category menu), and you will get a modal dialog where you can edit the text and image.

To list the categories in myCategoryGroup you do:

{% set categories = craft.category.group('myCategoryGroup') %}
{% for cat in categories %}
  {{cat.title}} - {{cat.image}} {# assuming image is the name of the image field #
{% endfor %}


Craft site has an article on How do I list the entries related to a category?.