darrinb

Build a better category list.
The Advanced Categories Widget is a powerful and flexible category widget for WordPress that makes building and displaying custom category lists a snap.
Organize your site in style with this highly customizable category widget!
Features
<ul>
<li>Build multiple lists!</li>
<li>Select which categories to display.</li>
<li>Choose how to order your categories: name, post count, etc.</li>
<li>Set the order for displaying the categories.</li>
<li>Supports category thumbnails.</li>
<li>Set and preview custom thumbnail sizes, or choose from your site’s registered image sizes.</li>
<li>Show a description of each category.</li>
<li>Set and preview the length of the description.</li>
<li>Select the list format to match your site’s markup. Choose from:<code>ol</code>, <code>ul</code>, or <code>div</code>.</li>
<li>Show post count.</li>
<li>Choose the default (basic) CSS styling or roll your own.</li>
</ul>
Developer Features
This widget was built not only with end-users in mind, but also plugin developers. Almost every aspect of this widget is extensible through filters and action hooks. You can even add your own form fields to the widget form!
<ul>
<li>Need to remove a field from the widget form? Not a problem! Every field is passed through its own filter for easy customization.</li>
<li>Need to modify the output of the widget? Easy! The output is passed through numerous filters, allowing you to customize the post list to meet your project’s requirements.</li>
<li>For a full list of action hooks and filters, please see the plugin documentation: http://darrinb.com/plugins/advanced-categories-widget</li>
</ul>

Advanced Categories Widget

A highly customizable categories widget for WordPress with thumbnails and descriptions.

codersantosh

AcmeIT

AddonsPress

If you have already familiar with the Categories widget, then You are already familiar with it too. 
It just lets you choose any taxonomy you have in your blog, for example, Tags, custom taxonomy ( Eg: Location, Product Categories, Product Tags etc… ).

Taxonomy Term Widget

Perfect

Add an advanced widget to your WordPress blog, like an extension of the Categories widget.

Advanced Categories Widget vs Taxonomy Term Widget