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To show only the title of posts on the site homepage, using the WordPress Default theme as an example, in the wp-content/themes/default/index.php file you will find code similar to this:

<h2><a href="<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>"><?php the_title(); ?></a></h2>
<small><?php the_time('F jS, Y') ?> <!-- by <?php the_author() ?> --></small>
<div class="entry">
<?php the_content('Read the rest of this entry »'); ?>
</div>

So, in this case, you would replace:

<?php the_content('Read the rest of this entry »'); ?>

with something like:

<?php
if (is_single()) {
the_content('Read the rest of this entry »');
}
else {//no content, nothing.
}
?>

This will cause only the post title to display on your homepage, but show the post content on the individual posts page. Note if your theme uses another template, such as single.php, to display individual posts, then this change may not affect your individual post pages.