Detect the taxonomy term(s) of a node from within a template file
As alluded to in the first answer, you should use a preprocess function, but probably for the node template.

For the node.tpl.php template (and all derivatives) the function is defined as

MYTHEME_preprocess_node(&$vars) {
}

and should appear your theme's template.php file.

You must clear the Drupal cache in order to reset the theme registry such that Drupal recognizes and executes this function.  

The $vars argument of the function is an array that contains all the template variables which subsequently appear in the tpl file ($vars['foo'] will be leveraged as &lt;?php print $foo; ?&gt; in the tpl file). 

This also includes the native $node object, which will allow you to inspect $node->taxonomy, and create any subsequent variables accordingly. You can leverage these in the tpl file.

The best way to inspect and discover the components of $vars is to use the Devel module's dpm() function, which nicely prints any variable to the theme's messages box.

So, enable devel, create the following in template.php, clear cache, refresh viewing a node, and work from there.

MYTHEME_preprocess_node(&$vars) {
  dpm($vars);
}

You may also be interested in my session on Drupal preprocess functions from Drupalcon Copenhagen or the Drupal handbook page.