This is the short story:
WordPress.com is a hosting platform that makes it easy for anyone to publish online. You don’t have to download software, pay for hosting, or manage a web server. WordPress.com has hundreds of themes, and includes the functionality of many plugins, but you can’t upload your own plugins or themes.
WordPress.org offers free software that you can install on a web server. You can upload and install themes and plugins, run ads, and edit the database.
But you should read http://en.support.wordpress.com/com-vs-org/ too.
If you’re interesting in bbPress you’ll want to go the WP.org route 🙂