A slug is a few words that describe a post or a page. Slugs are usually a URL friendly version of the post title (which has been automatically generated by WordPress), but a slug can be anything you like. Slugs are meant to be used with permalinks as they help describe what the content at the URL is.
Example post permalink: http://wordpress.org/development/2006/06/wordpress-203/
The slug for that post is "wordpress-203".
Slugs are also used for post types, categories, tags, etc. Like post and page slugs, this is usually for creating URL friendly versions of these items.