Does it show just to logged in admins, or does it show to everyone who has access to edit that post/page? I believe it is the later but that’s not what the original poster asked.
If you absolutely don’t want that link to show, you need to edit your theme files and look for the following code:
<?php edit_post_link('Edit this entry','','.'); ?>
Or something similar. The key part to look for is the edit_post_link
function – this is what adds the link. You will certainly find this in the single.php file, but you may find it in index.php and archive.php as well.
Be careful when you edit this. In the default theme the code is:
<?php } edit_post_link('Edit this entry','','.'); ?>
and to remove it you would need to change that line to:
<?php } ?>
or the theme will break.