WordPress.org

1. Upload post-comment-count.php to the /wp-content/plugins/ directory
2. Activate the plugin through the 'Plugins' menu in WordPress
3. Place <?php post_comment_count(); ?> in your templates. If used in "the loop" it will count the number of comments for the specific post, otherwise it will count the total number of comments in all posts

* You can use "total_comment_count()" and "total_approved_comment_count()" functions, if you need