If I register an Internet domain name (e.g. "mydomain.com") with you, who will own that domain name, you or I?

You will. We will register the domain name and will be listed as the administrative and technical contacts -- but you will be listed as the owner. You will hold all rights to transfer the domain name, sell it, etc. (Of course, if you also prefer to be listed as the administrative and technical contacts or you prefer to register it yourself, hey, that's fine with us. )