How do I only dis­play a list of cat­e­gories that have entries and hide emp­ty categories?

To display a list of categories that are in use (and make sure you are not displaying any categories that don't have content associated with them), you will need to take a couple of steps. Here's a barebones examples of the code we will need, and we will break it out into more detail below.

{% set entryIds = craft.entries.section('blog').ids() %}
{% set categories = craft.categories.relatedTo({ sourceElement: entryIds }).groupId(1).find() %}

{% for category in categories %}
<li><a href="{{ category.url }}">{{ category.title }}</a></li>
{% endfor %}

First, we want to grab all of the entry ids from the Section of content which we want to display categories for.

{% set entryIds = craft.entries.section('blog').ids() %}

Next, we will use the craft.categories tags relatedTo() method to only return categories that have a relationship with one of our entry ids.

{% set categories = craft.categories.relatedTo({ sourceElement: entryIds }).groupId(1).find() %}

Once we have our list of categories that are in use, we can loop through those categories and display them in our template:

<ul>
	{% for category in categories %}
  	<li><a href="{{ category.url }}">{{ category.title }}</a></li>
	{% endfor %}
</ul>