Template not being picked up?
(Assuming you are on Drupal 6, as it seems) you also need to copy the default node.tpl.php (e.g. the one from your base theme) in the same theme directory you are putting the override file (node-product.tpl.php) in order for the theme engine to pick it up:


  If you are implementing a suggestion, rename the file to the
  suggestion name it should have. Note that some suggestion files will
  not be recognized unless their base file is also present -- for
  instance, if you want to make a node content type suggestion file for
  your content type, you need to have node.tpl.php present in your theme
  directory order for your node-my_content_type.tpl.php file to be
  recognized.  (Beginners guide to overriding themable output)