WordPress Testers list provides you the news about latest beta version of WordPress. There are two ways of testing new beta releases, the first and hard way is backing-up your website database then downloading the zip file and then unzipping it and using it on your website via FTP. The easy way is doing the same thing from backend i.e. from admin area dashboard with WordPress Beta Tester plugin.
With WordPress BETA releases you can check new coming features of WordPress and test them locally on your computer or somewhere else. Example: For testing WordPress 3.6 you can download its beta .zip file here, upload them to a hosting or locally on your computer server, configure them just like you do with WordPress stable release installation and check it. Or you can replace WordPress core files on any running WP website and check BETA release, but remember its recommended not to run any BETA release on a production website. This tutorial explains you how you can easily upgrade WordPress to recent BETA releases by installing and activation a simple WordPress plugin.
Recently WordPress.Org made a beta release of WordPress 3.6. The software is still in development and its recommended not to run it on a production website, you can check its new features and look locally on your computer or somewhere else. For testing WordPress 3.6, try the WordPress Beta Tester plugin (you will want “bleeding edge nightlies”). or you can download its beta .zip file here.