WordPress.com is the commercially hosted side of WordPress. They provide a closed hosting system that they maintain, which limits certain capabilities on what you can do with your WordPress site but they take care of things like security and database maintenance, etc.
WordPress.org is where all the open source files for the WordPress installation and the self-help community are found. No websites are actually hosted here. It's just a repository for installation files, plugins, themes, and documentation pertaining to the WP program.
When you host at a place like GoDaddy, it's called a self-hosted WordPress. This means that you are responsible for the upkeep of all the things pertaining to your WordPress site. You have to pay for the server, you have to make sure the site is secure, you have to make sure maintenance is done, etc. This, in my opinion, is the best way to go if you want to be able to kit out your site the way you want.
So, to answer your question, you're not hosting your site on .org or .com if you're using GoDaddy. You have a self-hosted WordPress site.
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