What is "template_default"?

"template_default" is the master set of template files. Don't touch these (except when doing an upgrade).

"classic" is a sample custom template. You can create many custom templates. "classic" is just an example. Anything that's not in your custom template but exists in template_default will be automatically loaded from template_default for you. Thus, when you're building a custom template, all you need in your custom template folder is the files/folders that you've customized/changed/added.
If you need to customize the "specials" page for example, you would copy the template_default version of the specials template into your custom template's folders (using same folder structure as template_default), and make your changes to your custom copy. Same with images. and Stylesheets. and Buttons. etc.

See the other FAQs on the template system and on overrides for more information.