WordPress.org is the website where development of the WordPress software is centralized. The site includes the official Theme and Plugin repositories, the official WordPress project documentation wiki, and the official support forums. The site also includes the WordPress project bug tracking, and project development community contributor discussion areas for core development, WordPress UI, Theme review, accessibility, and translation.
WordPress.com is a hosted blogging service that runs on the WordPress platform. This means that you don't have to worry about technical issues (setup, upgrades, backups, scalability, ...), but you get less customization options: a limited number of plugins and themes, no ability to run custom PHP code, etc.
For more about the differences you can read "WordPress.com vs. WordPress.org".