It's where you refer someone to a product or service from your site using a special link that has a tracking code on it. Any time someone buys through your tracking link you earn a commission. Then the company pays you for any commissions you earn for the week, month, etc.
Say you had a site on big screen TV's and you wanted to recommend Best Buy's Samsung 60" Plasma TV. You would go to bestbuy.com and join their affiliate program (if they have one) and as an affiliate they provide you with link to embed into your site.
When your visitors click on the links from your site and go over to Best Buy and make a purchase, Best Buy pays you.
Thousands of sites have affiliate programs so once you decide what kind of products and services fit with your site, go to
www.associateprograms.com and search for programs you can join.