WordPress is a free, open-source publishing platform that can be used to easily set up a website or blog. In fact it's one of the most popular publishing platforms on the web–they claim to power more than 17% of the web.
Wordpress comes in two different versions: WordPress.org and WordPress.com (help page with side-by-side comparison here). The same software drives each version, but the key difference is where the software is hosted (.org is self-hosted but .com is hosted on Wordpress servers). Because the software itself is free and open-source, if you have your own server space then you can download the software from Wordpress.org (or install it through a system like Installatron) and load it onto your own server space, then customize it however you'd like. Wordpress.com is configured for people who do not have server space to host the software on, and it runs on a freemium model, where setting up a basic site is free but if you want certain kinds of customization, you have to pay for those upgraded features.
If you've registered a domain through Domain of One's Own @ Emory, then the Emory Writing Program is providing you with hosting space on a server for you to use. Therefore you can use the much more full-featured and customizable Wordpress.org on your own server space without paying any additional cost.