Since your code was moderated above, I cannot see your exact structure, so here is an example from a template I have:
<?php the_time(get_option('date_format')); ?>
<?php the_author_posts_link(); ?>
<?php comments_popup_link('No Comments', '1 Comment', '% Comments'); ?>
That lists, in order: the date | the author | comments
The comments_popup_link needs to be placed within the loop, which is probably why you are seeing it listed for the first post, but not on subsequent posts.
If you want to edit the text that shows for your comment links, you would edit the text within the comments_popup_link parenthesis - example:
<?php comments_popup_link('Leave a Comment', '1 Comment', '% Comments'); ?>
That will display if:
There are no comments yet: "Leave a Comment"
There is only one comment: "1 Comment"
There are several (like 10): "10 Comments"
Hope that helps.