You can do this by editing the your theme`s index.php file and adding the following code before the loop:
<?php query_posts('cat=xx); ?>
where xx is the ID number of the category you want to display, so your index.php file should look something like this:
<?php query_posts('cat=xx); ?>
<?php if (have_posts()) : while (have_posts()) : the_post(); ?>
......
You can also define the displayed category by its slug using
<?php query_posts('category=category-slug-name'); ?>
Also, you may want to check https://www.ads-software.com/support/topic/only-show-posts-from-a-particular-category-on-homepage where the explain everything in more detail, and even show how to limit your query.
