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There is a header on each category page
It would help if you don't state untrue things.

You are talking about your LEFT SIDEBAR, not the webpage.
Precise description of the problem is always helpful.

Move that h2 Catgeories /h2 inside the if statement to appear only when you have children cats.

Sorry... I've taken over tech duty for this blog, and WP is rather new to me. Thanks for the suggestion. Nonetheless, it is the if statement in particular I'm having difficulty with. Other than the main doc on conditional tags, where could I go for advice / reading on how to deal with this particular situation?

You could try this (which is exactly what moshu suggested you do):

<?php if ((is_category())||(is_single())) { ?>

<?php if($parent_category->cat_name == '') { ?>
<h2>Categories</h2>
      <?php wp_list_cats('child_of='.$this_category->cat_ID); ?> 

<?php } else { ?>

      <?php wp_list_cats('child_of='.$parent_category->cat_ID); ?> 

<?php } ?>

I tried moshu's suggestion, and the only change was that the horizontal bars above and below the "no categories" text disappeared.

<?php if($parent_category->cat_name == '') { ?>

Does this not just mean "If category name = anything"? If so, won't the rest of the code always try to display some sort of category children?

So you mean that is still displayed "No categories" in your sidebar and it still displayed the word "Categories"?

Try this:

<?php if ((is_category())||(is_single())) { ?>

<?php if($parent_category->cat_name == '') { ?>
<h2>Categories</h2>
      <?php wp_list_cats('child_of='.$this_category->cat_ID); ?> 

<?php } else { ?>

<p>TEST</p> 

<?php } ?>

Does the word TEST show up on your page?

Hi tsguitar, that is precisely what I mean. The "Categories" header and "No categories" text below it just won't go away!

I tried your code and the output stayed the same--the word TEST did not show up.

But does it show up on your other pages in other categories?

The TEST shows up in the sidebar of pages for the children of main categories (when children exist, of course), but it still shows "No Category" in the sidebar of category pages without children.

i'm having the same problem.
All i want is to display a menu that shows ONLY the children categories WHEN they exist in that parent category.
They already shows, with the code bellow:

<?php echo wp_list_categories('title_li=&exclude=1&use_desc_for_title=0&child_of='.$cat);?>

but in when a category dont have children, it reads: "No Category" all the time!