Categories and Parent Categories

In your sidebar (or Otto’s PHP Code Widget), use the template tag, wp_list_categories(), with something like this to show only ‘parent categories’:

<?php
wp_list_categories('depth=1');
?>

Then create a Category Template (e.g. call it category-dogs.php) and in that template use something as simple as:

<?php
/*
Template Name: category-dog Child categories
*/
?>

<?php get_header(); ?>

<div id="content" class="widecolumn">

<h2>Children of the category Dog:</h2>
<ul>
<?php
$current_cat = get_query_var('cat');
wp_list_categories('orderby=name&show_count=1&child_of=' . $current_cat);
?>
</ul>

</div>

<?php get_footer(); ?>

Related:
Stepping Into Template Tags
Stepping Into Templates
Template Hierarchy