A tag aware widget is one whose content can be controlled via a tag in a blog post. You can
read more about what you can do with tag aware widgets
here.
How to Use A Tag Aware Widget
When you are subscribing to a widget, a tag icon is displayed next to any field that is Tag Aware:

When you click the tag icon, you see this:

This is what the options do:
If you check just ‘Enable Tag-Awareness’

This allows you to use the <a rel="wbx-keywords> tag in your blog posts. For example, you
might make a tag to the word "tech" by blogging this:
<a rel="wbx-keywords" title="tech" />
This sends the keyword "tech" to any field of any widget to which you just
checked "Enable Tag-Awareness". You can see an example in a blog post
here.
If you additionally click ‘Support "rel" tags’:

Technorati and other tagging sites use the rel-tag
specification to tag content in a blog. We let these industry standard tags control widgets, too.
For example, you can tag the word "tech" by blogging this:
I like <a href="http://technorati.com/tag/tech" rel="tag">Technology</a>.
This sends the keyword "tech" to any field of any widget to which you checked ‘Support "rel" tags’.
If you additionally click ‘Named tag’:

This lets you send information to a specific parameter of a specific widget:
<a rel="wbx-keywords" name="srchTrms" title="tech" />
This will send the keyword "tech" to any widget that has a configuration parameter with a named tag "srchTrms".