Could be as simple as using the full function in the template file:
<?php comments_template( '', true ); ?>
Be sure you have a comments.php file in the theme folder.
And be sure you’re using the full loop:
<?php
require('/the/path/to/your/wp-blog-header.php');
?>
<?php
$posts = get_posts('numberposts=10&order=ASC&orderby=post_title');
foreach ($posts as $post) : setup_postdata( $post ); ?>
<?php the_date(); echo "<br />"; ?>
<?php the_title(); ?>
<?php the_content(); ?>
<?php comments_template( '', true ); ?>
<?php
endforeach;
?>
See Integrating WordPress with Your Website « WordPress Codex