On websites they often sell / link to other peoples websites or products.
For example if I run a blog about Marathon Running I may link to DVD's about running, a shoe store, clothing lines, maybe an iPhone app. For all of those services I'm linking too I want to get paid a commission on those referrals.
So all those other companies (dvd, shoe store, clothing store) would need an affiliate tracking solution to know what websites referred them traffic, and from that traffic what turned into sales, and from those sales how much of a % are they going to have to pay back to the original website that referred them.
Affiliate marketing is a popular way of making money on the internet.
And lots of the top blogs promote products, or services and get paid a commission on any sales made from people that come from their site. So the affiliate tracking software helps with that entire process.
Hope that helps!
Commission Junction is a popular site to find affiliates to link to, to make some money.