Using links, coupons, and promo codes.
When an affiliate link is clicked, a cookie file is dropped in your browser that allows Admitad to determine where the order came from and which publisher is entitled to the reward. Coupons also usually include a publisher's mark in the link.
As for the promotional codes, things are a bit more complicated: they get scattered all over the Internet, and establishing who exactly brought in the client is not always possible. Admitad advertisers solve this problem by offering personal promo codes all sales by which are attributed to the specific publisher.