Someone who I am neither following nor is following me appeared on my home page on twitter, why?
He is followed by someone who you follow, and his tweet was retweeted by the person you follow. That's why his tweet appeared on your timeline. For more about retweeting, you may see this page.
In addition to one of your friends pushing the official "retweet" button, Twitter is rolling out a feature that brings ads to your feed as a normal post. Here is an informative news article so you can figure out if this is happening to you. It sounds like it only happens if you're following some kind of brand, famous person associated with a brand, or spokesperson - so if you follow your 20 close friends and no one else you'll never see one. Or something. They're not very clear.