The Beginner's Drupal Administration Series
The 'Appearance' admin section is the place where you enable, add, and configure the themes available on your site.
Appearance main page
If you click directly on the 'Appearance' link (without hovering to the expanded child links), you will see the page lists all the themes you have downloaded to your site (you can do this by either 1) downloading and unzipping them to your /themes directory or 2) using the 'Install new theme' link below).
The 'Enabled themes' section lists the themes you have enabled on your site - you can set any of these as the default theme to apply it to your site, and you will be able to configure any of your enabled themes in the 'Settings' section. The 'Disabled themes' section lists the remainder of the themes you have downloaded on your site, but have not enabled. You cannot set these as your site's current theme or configure their settings until you actually enable them for usage.
The 'Administration theme' section allows you to choose which of your enabled themes to use when viewing administration pages (such as all the pages discussed in this administration series). I strongly suggest that you use an administration theme that is a stable, well-used theme (such as the theme Seven, which is enabled by default). If you use a custom theme for your admin pages that is the same theme you use for your site's public pages, you may end up messing it up during development and have problems viewing your admin pages to navigate away from the problematic theme! So choose something stable, that you do not modify for your admin pages.
Install new theme
In this section, you can download new themes for your site. Go to a theme's drupal page and scroll to the 'Downloads' section.
Copy the link location for the latest recommended tar.gz file. (If you're using IE, it will be called 'Copy shortcut'). Paste the saved link into the 'Install from a URL' section, and Drupal will download and extract the file and place it in your /sites/all/themes folder. It will not be enabled at this point, but you can go back to the 'Appearance' main page and enable the new theme!
This section lists pending updates for your enabled themes, modules and drupal core. For module updates, you can use the admin interface directly to apply updates. However, for drupal core updates, this functionality is not yet available for Drupal 7, and you will have to do updates manually...
This section lets you configure settings that are global (for all your enabled themes) or specific to each of your enabled themes. Remember, you will not be able to configure themes that you have on your site, which are not enabled.
The global settings allow you to check whether you want your themes to display your site's logo, site name, site slogan, whether to display user pictures in comments, and other options. If you click on a specific theme, however, you may get a wider range of individual settings. Themes implement admin UI options independently and you may get themes that do not have any additional options from what you see in the global settings, to themes that have a huge array of additional options to govern their appearance. Click on each of your enabled themes to see what configuration settings they may include!