If you want to make some more fundamental changes to a template, Joomla! allows you to edit the HTML. There's an HTML editor available in the Joomla! backend, just like the CSS editor screen. If you want, you can thoroughly change the template code. You can add, change, delete, or move any existing page element—columns, header, and footer, whatever you like. Of course, you should only do this if you know your way around in HTML; don't risk messing up the site layout.
Even if you're not aiming to immerse yourself in the nitty gritty of HTML, being able to change the template HTML directly in Joomla! is still useful. It allows you to change or remove unwanted items that are sometimes "hard coded" (that is, a fixed part of the HTML code) into the template, such as a footer text or copyright notice.