What is to be an affiliate or a Publisher

In the internet, both mean the same. This is that you (the affiliate or publisher) promote a product or a service whether any via through internet or any other via, but you have to direct a potential customer to your affiliate link in the web (watch for the explanation below). Then, depending of the affiliate program you will earn certain amount of money as could be the following:

• Just for the potential customer visit the link.
• For the potential customer fill a form that could be of only 1 blank as for example the e-mail adress, as also could be a lot of more blanks.
• For the potential customer actually purchase the product or service your are promoting. In this case you could earn a fixed amount of money per sale or a direct commission of it.
  

The first two options, in which you earn money for creating a lead, are known as CPA (Cost per action) programs.
What is an affiliate link?
As referred above, this is the "sub-site" of the main site of the program you are publishing through which the system will acknowledge that the potential customer is coming from you. In the most simple way this could be like this:
Main site: http://www.their-site.com
Your link or "sub-site": http://www.their-site.com/your-alias
Most of the times, and depending of the specific program, they will install an expirable or non-expirable cookie in the system of the potential customer, in order to recognize the customer and the sender (you) in case the customer go back to the page later to make another purchase or to purchase for the first time in case he was not sure to make it the first time he visited.

What is a cookie?
A cookie is usually a tiny text file that installs in a system which is used to recognize some patterns when the web navigates again for the same site. They use to be harmless and can be deleted easily from the control panel of the computer, but doing this would incurr in a less efficient navigation.