The merchant is the seller or the manufacturer of the product or service. The party that wants to increase sales and leads.
The publisher is the advertiser. Examples of publishers would be newspaper websites, cash back sites and niche specific blogs. The idea is that these websites get the warm traffic, they then send that traffic through to a merchant's website and hopefully a sale will be made. The publisher then receives a commission for the sale.
The person buying the product or service.
The network is the framework of the whole process. They onboard suitable and trustworthy merchants and publishers and handle all the payments. They are sort of like PayPal in terms of being a middleman between all parties.