According to Google, An attribution model is the rule, or set of rules, that determines how credit for sales and conversions is assigned to touchpoints in conversion paths.
There are many problems associated with attribution models. For instance, how do you blend long-term effects (a user who see a same TV ad for a few months) with short-term actions (clicking on a Google ad and making a purchase within 24 hours). How do you assign the proper weight to each channel: the TV ad, versus the Google click, or versus doing a Google search and finding/purchasing your product without burning your advertising budget.
You can find answers in this article or here. The problem is also discussed in Granville's book page 248. In this short note however, we discuss a rather specific but fundamental issue: given a sale acquired via paid advertising, what is the chance that the sales would have occurred naturally, organically, without advertising at all? If the chance is 50%, it needs to be factored in in your cost of acquisition; it makes your advertising effort two times more expensive than you believe.
Here is a simple example: Twitter ads
You purchase advertising on Twitter to acquire new followers. If you do not advertise, you get 120 free new followers each day. If you advertise, you get 80 new followers from paid ads, and 80 free new followers each day. In short, you think you get 80 new followers from your paid campaign each day, but actually you only get 40, since you are loosing a big chunk of free followers who would have followed you anyway, even without your paid ad.
However, the followers that you get from paid ads are more targeted, for instance US-based only. So maybe while technically you only get 40 new followers (above your organic baseline) each day, these paid followers could be worth more, they could be worth (say) 60 followers, but probably not as much as 80 followers.
You need to take this into account when computing your cost of user/customer acquisition. It has a big impact on your bottom line. In the case of Twitter, Google AdWords, Facebook or other channels, it's pretty easy to make the computation by stopping your campaign for a few days and see the impact.