Any person can choose to be a CA. At Pediatric Heart Center, we decided to be our own CA, so we maintain a root certificate and private key, and for every certificate that we sign, we update a database with the serial number of the new certificate and other vital details about the certificate that was signed. We then are able to provide these certificates to employees for secure communications, and because all our systems recognize and trust the self-signed root, all the child certificates that we create are automatically accepted by our employees' systems.
Where the network of trust becomes a little bit more hairy is for those members of the general public who must choose to recognize our root certificate and then decide how much trust to provide to it. For our customers, we are able to give them a little education on what they can expect and how to proceed with providing "Trust" to our root certificate on their various systems.
Each Web Browser comes with a set of root certificates already installed and "pre-trusted". These Certificate Authorities have been well established and are no longer questioned by the public at large.
If you are looking to be among these root certificates, which are pre-installed and pre-trusted, your best bet is to contact the browser manufacturers, and if possible have your credentials and root CA certificate authorized by a central internet standards board.
Mr. Deters suggested that ICANN was a good place to start with his somewhat sarcastic comment on your original post. This might actually be a good place to start. Be ready to provide documentation about your signing practices and be open to random inspection, because these are both required of all Certificate Authorities, even private Certificate Authorities like ours at PHC.