The Template Editor changes a template in your browser, without FTP. Skins have it on the Templates tab (ACP > Skins > (skin) > Templates). Modules have one too, on ACP > Modules > (module) > Templates.
Your changes are saved in the database. The template file isn't changed, so you can always go back to it.
Edit a clone, not a Marketplace skin. See Cloning a Skin.
A saved change isn't used until you activate it. Back on the list, the notice says "The highlighted customized templates are NOT ACTIVE!", and the Active box of your template is highlighted.
Check Active, and click Save Changes at the bottom of the list. Your change is now on the site.