Anyone can buy a domain name and point it at any IP address on the Internet. Using redirect method, the domain name can point to your domain, in which case the visitor's browser shows your address on arrival. If it points to
http://xxx.xxx.xxx.xxx IP address, your IP address would appear in the visitor's browser, unless there is some redirect code on your server to redirect permanent to your domain.
Using the frame method, a fremeset is hosted by the domain registrar, with your pages inside the frame, and the offending domain name always showing in the visitor's browser. You can force your pages out of a frame using Javascript or server-side code.
I point a subdomain of one of my hosted sites to another site that does not belong to me, and when I do a check of my site using Google's webmaster tools, I see that there is always an error indicating that the other site has performed a redirect permanent that advises search engines of the correct address. I am basically doing a TinyURL service from my site in making a short URL for a resource on the other site.
Perhaps this information may point you in the direction you need to take to re-establish your domain's rule over its territory.
Also, since I am on shared hosting, I have no experience with a dedicated IP, and what precautions can be taken to ensure that DNS records are correctly established for your domain name.
EDIT:
Additionally, Google can be told which version of your domain you want it to track,
http://example.com or
http://www.example.com. If your site can be opened either way, you need to redirect permanent away from the way you don't want, and also advise Google of your preference in its webmaster tools dashboard. You can also redirect permanent away from the offending domain and advise Google of your preference.
Having a site's content displayed in both www and non-www splits your site's popularity with Google between two identical sites, with penalty added for having identical sites, so your site's rank is hurt.