I am not sure if your question is starting an affiliate program or promoting affiliate products, so I'll answer both.
If you want to promote a product that you own you will need an affiliate management script that is installed on your website or you will need to list your product on an affiliate network like ClickBank or CJ. Doing the later will cost you fees, but will get your product out on the market faster.
If you want to run it yourself you can check out these three programs:
1. Post Affiliate Pro
2. iDev Affiliate
3. JROX
Now here is how you would make money promoting affiliate products.
1. Find a niche and a product: You can search ClickBank, CJ, Linkshare, PayDotCom or any other affiliate network that offers commissions for sales.
2. Register a domain with the product name in it.
3. Install a Wordpress blog on your domain.
4. Put some banners on it that are provided by your affiliate program.
5. Write 3-5 blog posts about your product and anything related to the niche.
6. Write 10-12 articles about things related to the niche or a review of the product and link them back to the blog and also use social bookmark websites to bookmark the articles. Submit the articles to EzineArticles.com and a few other directories.
7. Place one new post at least once a week on your main blog.
8. Continue building backlinks by creating webpages with Squidoo, HubPages, Google Pages, Netscape, Digg or any other high PR site that lets your submit content and link them back to your blog.
9. You should start to see very targeted traffic in a couple of weeks and start to see sales come in.
10. Just keep updating the blog every now and then and go on to repeat the process with a new product and niche or choose another product in the same niche.
If you do this consistently for a year you can make some real good money off of free traffic and affiliate programs.