WordPress History
WordPress was started in 2003 when a developer named Matt Mullenweg took an existing platform and expanded the features to make it better. In 2004 WordPress got GNU licensing to become an open source platform and began to evolve into a blogging platform. The ability to add plugins was also added and in 2008 pre-made themes were included.
WordPress.org Entity
Wordpress.org is also known as self hosted WordPress. This means that the user installs WordPress on their own (or a hosting company's) server and sets it all up. A self hosted WordPress site allows you to install whatever themes and plugins you wish. The user is responsible for making sure that the WordPress core, themes, and plugins are updated.
WordPress.com Entity
Wordpress.com is a managed hosting platform for WordPress. There is no installation involved. A user simply signs up for an account and starts to build their site. On Wordpress.com, installation of themes and plugins is more limited. WordPress.com handles all updates for themes, plugins, and the WordPress core.
Summary of the Documentation
While both Wordpress.com and WordPress.org offer the WordPress platform with tons of options for customizing the look of your site, there are pros and cons to each option. The .org/self-hosted option allows for more options and customization and control of your site but is more hands on as a user since you need to install it, pick plugins & themes, and keep them updated. The .com option is likely easier for a new or non-technial user as it takes the installation aspect out and allows the host to manage the updates and installation. But it also reduces some of the customizabiity of the platform.