An affiliate is an individual or company which sends potential customers to a product, or someone who receives a commission on a sale.
In general, you have to sign up to become an affiliate of a particular vendor, product, or service. Once approved, you’re given special tracking links that, when clicked, indicate a customer has come to the product website through your link. Should they then purchase the product (or in some cases, any product), you’re given a percentage of the sale. Percentages range from 5% to 50% or more, depending on the product and affiliate program.
Naturally, affiliate programs are meant as an incentive to promote a particular product or sales venues.
For the record, while Ask Leo! does indeed use affiliate links when available, the availability or characteristics of an affiliate program for a specific product does not affect whether that product is mentioned or recommended. More on the topic specific to Ask Leo! here: Product Reviews, Recommendations and Affiliate Links Disclosure.