How Do Websites Recognize you?

There's a certain website, where I purchased a CD and In return I was given something else. The CD was only 4 dollars, but what you get in return is 20 dollars. So I wanted to do this offer again. I changed my ip and went on my laptop. Basically, I had a different ip and computer, how could the website possibly realize who I was? And oddly enough, It didn't let me do the offer again. How the hell do certain websites recognize who you are, even after changing your computer and ip?

name, address? don't know anything else...

Typically there are a few ways sites can recognize you. IP address, browser and version, operating system/version. Some even save a cookie on your computer or use a constant session meaning that the session ID as well as several other factors are stored in a database and resurrected when you log back in.