Technically, it isn't an application as much as php and css code, that is distributed as a open source blogging/publishing platform.
WordPress.org is where you get it. You'll need to download it, uncompress the files, and set up the config file. You'll need to set up a MySQL database on DreamHost and put the MySQL account info in the WordPress config file. Then you upload (FTP) all of the WordPress files and follow the installation info. For the technically inclined, this is pretty easy, and it gives you complete control over WordPress. WordPress.com does all of this for you, which is why they charge for the service.
I believe that DreamHost offers a "one-click" install of the software for the less technically inclined. This option is much easier to install, but then you are limited to how DreamHost has WordPress configured. Since I am not an expert in their setup, you may have limited choices with this option (as in you might lose the ability to edit CSS, etc.).
For more info, you should read DreamHost's wiki on WordPress:
WordPress makes its money off of donations and its parent company, Automatic, which offers paid services, such as WordPress.com. Because it is open-source, it is "owned" by everyone.
Think of WordPress.com as the commercial venture, and WordPress.org as the community-driven, open-source platform for all.
If you want to learn more, here is a good article, and if you enjoy it, check out the rest of the site for tutorials and advice: