You need to echo get_post_meta()
, like this:
<div class="button">
<a href="https://wordpress.stackexchange.com/questions/48542/<?php echo get_post_meta($post->ID,"Button', true); ?>">
<img src="<?php echo get_template_directory_uri() . '/images/button.png'; ?>" alt="link" />
</a>
</div>