@JoeTaxpayer's answer is good sense. However, if you just want to know the mechanics, it's simply a matter of logging in to the web site of the card you want to transfer a balance too. If it's a 0% interest offer, they most likely will have sent you email or snail mail with a link, otherwise just search the site for "balance transfer offer" - they're usually prominently displayed when you log in.
Once you have the link, and have read/understood the terms, you just enter the number of the credit card you want to transfer the balance from, and the amount you want to transfer, and click on the "confirm" (or whatever the particular site calls it) button. You should get a confirmation email within a few hours, saying your transfer is in process. Within a few days, you should get another email from the TO card, saying the transfer was successful, and one from the FROM card saying a payment was made.
Note: I'm assuming US cards here. Other countries might differ.