craigsimps

Widgetized themes often include support for adding a featured page or post, but often there’s no easy way to feature a category from your blog.
With Featured Category Widget you can easily featured a blog category (or a category from a custom taxonomy, like products), add a custom image and optionally show the category description and a list of sub categories.
Features:
<ul>
<li>Add custom image and set display size.</li>
<li>Optionally show category description and/or subcategories.</li>
<li>Supports custom taxonomies.</li>
<li>Developer friendly hooks and filters throughout.</li>
<li>Translated into French, Spanish, German.</li>
</ul>

Featured Category Widget

Showcase a specific category with ease, including setting a featured image and listing child categories.

tepelstreel

The Featured Category Widget is mainly designed because there were people for whom the Featured Post Widget was not enough. They wanted to put a category of their blog in the highlight. 
If there is a post thumbnail, it will be displayed above the headline of the post. If there is no thumbnail, the first picture of the post is taken. You can set the size for the thumbnail or just take the standard from your options. Decide yourself, whether you want to show the excerpt, saved with your post or just the first three sentences or the first twenty words of the post. Style the widget individually, ready.
The Featured Category was tested up to WP 4.5. It should work with versions down to 2.9 but was never tested on those.

The Featured Category Widget is basically a Featured Post Widget for a category.

Featured Category Widget vs Featured Category Widget