How do I use a Twig template to generate Email content from inside a plugin I'm making?
Thanks to Brad for providing the bulk of the answer. In more detail the line needing to be changed was this:

$notificationEmail -&gt; htmlBody = craft()-&gt;templates-&gt;render('PLUGIN-FOLDER-NAME/TEMPLATE-NAME', ['theUser' =&gt; $user]);


I could then use {{ theUser }} (Which is a UserModel) in the TEMPLATE-NAME template to generate everything I needed.

So the full solution, to send an Email from a plugin, but using a Twig template to generate the HTML email content was:

$myEmail             = new EmailModel();
$myEmail -&gt; toEmail  = $user-&gt;email;
$myEmail -&gt; subject  = "Subject here";
$myEmail -&gt; body     = "PLAIN TEXT CONTENT";
$myEmail -&gt; htmlBody = craft()-&gt;templates-&gt;render('PLUGIN-FOLDER-NAME/TEMPLATE-NAME', ['theUser' =&gt; $user]);
craft()-&gt;email-&gt;sendEmail($myEmail);


Where $user was a variable containing a UserModel which I'd already got from previous code in the plugin I was writing.

https://craftcms.com/classreference/services/TemplatesService#render-detail for more detail (but for which I wish there were example code of how to use, and not just a dictionary).