Last week Star Wars: The Old Republic began its free character transfer service to help balance out server populations throughout the game. Players can transfer their characters from select origin shards, and relocate them to select destination shards. In order to be eligible for a free character transfer a player’s account must not be banned, and the character selected may not be a guild leader. Guilds are not eligible for transfer.
To start the character transfer process, login to your account, click My Account and then click Character Transfers in the left hand menu. From there you will be able to select Start a Character Transfer and see if you have any characters eligible for transfer to selected servers. You can check out the official Character Transfer FAQ for additional information.
There is quite a large list of both NA and EU servers eligible for transfer right now and you can head over to SWTOR.com/transfer to see if your server is on the list. Good luck in your search for a new home!