For more comprehensive reading:
To answer your question, first thing I would do is copy your entire idex.php file in a text editor, and save it as home.php. That way, depending on your theme, you are only making changes to the home page, not all pages that require the index.php template
(More indepth reading- http://codex.wordpress.org/Template_Hierarchy)
Once in home.php look for this:
<?php if (have_posts()) : ?>
<?php while (have_posts()) : the_post(); ?>
That is where the loop starts.
All you want then is this section
<div class="post" id="post-<?php the_ID(); ?>">
<h2><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title(); ?>"><?php the_title(); ?></a></h2>
Delete - or better yet, comment out by adding <!-- at the beginning of what you don't want, and --> at the end of what you don't want to use.
Depending on the theme, there may be some conditionals afterward, but you want to stop the deleting or commenting when you get to this:
<?php endwhile; ?>
See if that gets you what you want, and if not, let us know.