A short definition of affiliate marketing in my words is a person or company gets paid by a business to promote products or services.
The long definition would just go into a bit more detail. There are really two main parties in affiliate marketing with another party that is also involved largely. First, you have the advertiser who is looking for people to buy their product or for people who are interested in a certain service, such as attorneys looking for new cases, car insurance companies looking for new clients, etc.
Then, you would have the advertiser, who is the one looking to get their product promoted for a price. They enlist affiliates to go out and market their product alongside their own marketing efforts as well. This is often a cost-efficient method, as you pay for results instead of a typical advertising campaign.
Often, there is something called an affiliate network, which often works with advertisers to facilitate a large number of offers for the affiliates. In this case, the affiliate network takes care of collecting the payment from the advertiser so that you don’t have to. Often, you can get lots of larger advertisers when you belong to an affiliate network because they have the power to offer large amounts of traffic to these companies.
To sum it up, an affiliate is responsible for all advertising of the product/service, including costs. The affiliate network provides the offers for the affiliates and pays the affiliate out of the money they collect. Finally, the advertiser is interested in getting a 3rd party to promote their product, thus needing affiliates and affiliate networks.