Most of the editing you'll be doing will be in the style.css file. The images will not be placed in the (X)HTML code but will be set as a background with CSS. Since you have not mentioned which theme you're using nor posted the code from the header.php, I'll use the code from the "Default" theme of WP.
Open the header.php and find:
<h1><a href="<?php echo get_option('home'); ?>/"><?php bloginfo('name'); ?></a></h1>
<div class="description"><?php bloginfo('description'); ?></div>
<div id="headerimg"> to
<div id="headerimg" class="cat_<?php single_cat_title(); ?>">
This means if you're on the category page named "cars" the output of this is will be
<div id="headerimg" class="cat_cars">
(note this will only work on the category pages not for "home" and "about" because those are "Pages" for Pages you will need to create Page templates.)
add for each category the corresponding code. For the cars category it would look something like this:
Naturally you may need place the image named cars.jpg in a folder named images and adjust the width and height to the actual dimensions of the image.