Wordpress.org is the home of the open source WordPress software. It is code you can download and do as you wish with. You can take that code, upload it to your own web hosting, and set up a blog you have complete control of.
Wordpress.com is a blog hosting service. It's a free web host where instead of getting FTP access to upload your own files, you get a login to a WordPress blog set up for you at sign up, on their servers. You have no direct access to or control of the site, so you can't install your own plugins or write your own code.