For the one time I did it with a domain name registered with Network Solutions, the new registrant initiates a request for transfer, paying a US$3 fee in the process.
The original registrant gets the request via email pointing to a URL to approve the transfer. You then select confirmation or denial.
If the original registrant does nothing in two weeks, the transfer the aborted and the US$3 won't be refunded BTW.
The procedure should be quite similar for other domain name registrars.