Here is a brief rundown on the difference between WordPress.org and WordPress.com:
WordPress.org, also known as self-hosted WordPress, is open-source WordPress software that you can install on your own web host to create a website that is wholly yours.
WordPress.com is a for-profit, paid service that is based upon the WordPress.org software. It is easy to use, but lacks much of the flexibility of self-hosted WordPress.
When people refer to "WordPress", they usually refer to the self-hosted version available at WordPress.org. It is almost always best to use self-hosted WordPress.org if you wish to truly own your website.