Affiliate software is any installed or hosted management solution that helps an online merchant, who has an affiliate program, track the sales and commissions generated by their affiliates.

There are different types of affiliate software solutions:

• Install yourself on your server - custom or purchased off the shelf - e.g. iDevAffiliate or Post Affiliate Pro and many other affiliate software packages
  
• Hosted by a third party and sold as a service but not as part of a network - e.g. AffPlanet
  
• Hosted as part of a network, where you sign up as an advertiser on the network and the network has more than one advertiser on it - e.g. Shareasale
• Included as a module in a business site builder package or shopping cart - e.g. 1shoppingcart
  

Here is a good definition of affiliate software on Google

Still not sure why you need this sort of solution?

This 2 minute tutorial I made gives a brief overview of how affiliate marketing works, and you'll start to see why the advertiser managing the affiliate program needs to accurately track who referred who: