Using the WordPress Default Theme for discussion, to do this you would change the wp-content/themes/default/archive.php file and delete these lines:
<div class="entry">
<?php the_content() ?>
</div>
<p class="postmetadata"><?php the_tags('Tags: ', ', ', ''); ?> Posted in <?php the_category(', ') ?> | <?php edit_post_link('Edit', '', ' | '); ?> <?php comments_popup_link('No Comments »', '1 Comment »', '% Comments »'); ?>
</div>
With the help of the Template Hierarchy article, I determined what Template displayed the category archives, then I edited that template and deleted reference to template tag, the_content(), which displays the content of posts. I also deleted the code that displayed categories and tags. What that left was a line of code that displayed the title of posts in the form of links. That line is:
<h3 id="post-<?php the_ID(); ?>"><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h3>
Thanks very much I will give that a go…