The WordPress theme check is a great plugin. For those who haven't seen it it's a way to check that you theme covers all of the functionality required to be added to the WordPress.org theme directory. I have been using this plugin to help me test the themes that I am creating on Pro Theme Design.
It covers a surprisingly wide array of tests, making sure that you don't have any errors, and that you support all the most important elements of WordPress (to provide maximum bang for buck).
One of the features it checks for is use of the $content_width variable - and this isn't something I have ever seen used or mentioned, so I thought I would do a bit of research into it.
So, it turns out, that the usage is simple. The variable is a global variable that is used to set the width of the content on the site. The only thing this affects is the images uploaded to the image uploader, and videos embedded with the video shortcodes. Setting the variable correctly means that the images will have a resized version that fits perfectly within your content - so it's definitely a nice thing to include.
The recommended implementation is to add the following to your functions.php file in the theme root.
if ( ! isset( $content_width ) ) $content_width = 900;
Personally I would like to see this variable filterable so that I don't have to add some random floating code in my functions.php
If you want to try the WordPress Theme Check plugin for yourself then you can download it from WordPress.org.