I've managed my company's own x509 DV certificate, so I'll tell you based on my own experience.
Basically, you privide the CA with an administration email address in your certficate signing request (CSR), typically webmaster@ or admin@ <your-domain-name>. Then, CA sends you an email, and you follow the links or instructions in the mail and approve your own CSR.
OV and EV might involve other procedures.