To put it in simple terms:
[*] WordPress is the name of the software. Not “wordpress.org” or “wordpress.com” — those are both web sites.
[*] wordpress.org is the web site that you can download WordPress from, along with WordPress themes and plugins. It’s also where the WordPress developers hang out.
[*] wordpress.com is a company that hosts some WordPress blogs. They have a free starting option, but various upgrades that you’re likely to want (e.g, removing ads, using your own domain name, customizing the theme) cost money.
While it’s possible to point your DreamHost domain to your existing blog at wordpress.com, you will need to pay them for that upgrade. Since you already have a hosting plan with DreamHost, you’d be best off moving your blog over here like Ipstenu described.