Process might vary slightly, but generally:
1) Ring new credit card company when you have received your new credit card.
2) Tell them how much you want to transfer and give them the number of the old credit card.
In essence, that's it. If you've completed this information on the application form, it might be done for you when the card is set up.
Once the balance has been transferred, remember to cancel the old card. If you've paid the credit card levy to the old company, get a letter from them confirming this and give it to the new company. This way you won't pay it twice in the same year.