WordPress is a Content Management System (CMS).
In plain terms, it lets you easily manage your online content. It initially became popular as a blogging tool but has evolved considerably since. Almost any type of website that you can imagine can be built on WordPress.
More websites are powered by WordPress than any other Content Management System.
WordPress.com is an online blogging platform. You do not need web hosting to run your website/blog from WordPress.com but you are limited in scope with what you can do. It is mostly suited as a blogging platform.
WordPress.org provides the downloadable Open Source WordPress CMS. It is the same software that powers WordPress.com and is available under Open Source license. You have full freedom to edit the source code and modify it to suit your requirements. You’d however need your own hosting to run the website.