Place the following in sidebar.php or whatever file your theme uses to display the sidebar,
<?php global $post;
$categories = get_the_category();
foreach ($categories as $category) :
?>
<h3>More News From This Category</h3>
<ul>
<?php
$posts = get_posts('numberposts=20&category='. $category->term_id);
foreach($posts as $post) :
?>
<li><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></li>
<?php endforeach; ?>
<li><strong><a href="<?php echo get_category_link($category->term_id);?>" title="View all posts filed under <?php echo $category->name; ?>">ARCHIVE FOR '<?php echo $category->name; ?>' CATEGORY »</a></strong></li>
<?php endforeach; ?>
</ul>
This will display the titles of the most 20 recent entries of whatever category the post displayed on single.php belongs to.
In other words, someone arrives at the home page of your blog. They start reading a post about cars that belongs to the Automobile category. The click the title (or read more link or whatever) and arrive at the single post page (single.php) to finish reading the article. In the sidebar will be a list of the 20 most recent posts belonging to the Automobile category.
To change the number of titles displayed simply alter numberposts=20