The two main parties involved in the affiliate relationship are the merchant (sometimes also called “advertiser”), and the affiliate (sometimes called “publisher”). There are different ways to run, manage and promote affiliate programs, which involve more parties in the relationship, but the two main participants (without which the existence of the very marketing channel would’ve not been possible) are: (a) the party that has the product (or service), and (b) the party that knows how to sell it.
The best way to find suitable brands to promote is by simply using a search engine using: '[Brand] + Affiliate Program'. Some companies run in-house affiliate programs however, this is a very specialist area. Therefore, most companies opt to employ an 'affiliate network' which has already built a large base of affiliates and gained years of experience in running successful programs for clients.
Good comment Jason, at some time in the future Amazon may decide they have so much market share they don't need affiliates anyway. I mean, if you're just sending them people who are already Amazon customers there's not so much benefit there for them. Or they may decide to only work with select HIGH QUALITY affiliates and the average "affiliate site" owner will not be chosen.
You should also make sure you aren't competing with your own affiliates for eyeballs. Any marketing channels you're using, such as search engines, content sites or e-mail lists, should be off limits to your affiliates. Put marketing restrictions into your affiliate agreement and notify partners immediately. It's your program--you set the rules. Or, if you prefer, you can let your affiliates run the majority of your internet marketing.