At long last, Bioware have begun the process of shuffling their player population to better manage empty and full servers.
On a post on the official SWTOR site, the Bioware team explain that "the character transfer service will be made available gradually, starting with a small number of the lowest population servers and adding more until all servers eligible for transfer are made available in the service."
Currently, server transfers are free, but the help documentation leaves the option for paid for character transfer open. "Future versions of the Character Transfer Service may allow more options in your choice of destination servers and may require a transfer fee."
The list of servers available for transfer is here. You can begin your server transfer via the official SWTOR site.