WordPress has a cool feature. When you’re logged in as a user who is either an Editor or an Administrator, your pages will show a link “Edit this entry” right on the page. clicking the link will enable you to make a modification to the page.
What could be easier? If you don’t want the link to show, don’t be logged in as user with the role of Editor or Administrator. It’s really that simple.
OK. What’s plan B? If you are really serious about eliminating the link entirely, you’ll need to modify you WordPress theme template.
For instance in the WordPress Default theme’s wp-content/themes/default/index.php this code presents that type of link:
<?php edit_post_link('Edit', '', ' | '); ?>
You’ll need to carefully remove that piece of code.