I bought a Gateway and went to the many times provided link and entered all my information. In the end thay gave me a tracking number and sent me an e-mail confirmation. Some manufacturers require and receipt and money while others do not, Gateway didn't require either. 
They will ask for your S/N number on your computer, make sure you have this handy, it will then display automatically (for mine anyways) the model number and whether or not it qualifies. Its a quick process, they just need to know which computer you bought, the S/N number, and the date. You will then enter your name, address, phone number, and e-mail address. In the end they will send you an e-mail.
If you have to pay anything they will ask for your credit card number at the bottom of the one screen, mainly for shipping purposes.
Just click on one of the many links provided and then select which manufacturer you bought from, just fill everything out. I think there was 5 or 6 steps, took me a few minutes to fill it out.
Since you purchased your computer since June 26 of this year, go to here. Click the link for your computer's manufacturer, fill out their form, and they'll mail you a disk you can install Windows 7 from. Take note some manufacturers require you to send a copy of your receipt, and some require you to pay for shipping.
