How to get the URL for an image thumbnail
You need to use function image_style_url($style_name, $path) to obtain proper path in Drupal 7.20 and later. $path is a path to an original file. Preferably output normal path, and make it a thumbnail in proper .tpl.php file.

If you need to do it in place that does not accept php functions and cannot alter it in template, you can consider disabling itok token. To do so, add following line to your settings.php

$conf['image_allow_insecure_derivatives'] = TRUE;


Then you can simply add styles/thumbnail/public/ like Kevin says, and use path like example.com/sites/default/files/styles/thumbnail/public/myimage.jpg to get thumbnail of example.com/sites/default/files/myimage.jpg.