Hey Vanessa! I'm presenting the Move Your Blog to WordPress session at BlogHer '13 and I can help you out! WordPress.org is run by the WordPress foundation and they are the ones who guide the development of the full WordPress platform. You have complete control and can monetize, add functionality and do anything you want to it. Most importantly - you own your content.
One of the original developers of WordPress took the platform and created a pared-down version of it to function as a blogging platform. That's WordPress.com. The dashboard looks the same as the full version, but there are a lot of things missing like plugins and theme compatibility.
Here is a full post on the differences that goes into a bit more detail and has an infographic comparison: http://www.boostinteractivemedia.com/the-difference-between-wordpress-co...
Drop by my session on Friday morning if you get a chance! I'll be going through all of the major steps and what to watch out for during your move.