The Attention Economy is a marketplace where consumers agree to receive services in exchange for their attention. Examples include personalized news, personalized search, alerts and recommendations to buy. Note that the Attention Economy is different from the tradional meaning of an economy, because it isn't about buying and selling - although ultimately those things may occur.
News feeds illustrate the point well, since they ask for consumers attention in exchange for the opportunity to show them advertising. Search engines also show ads (asking for consumers attention) in exchange for helping users find answers online (a service provided for free in exchange for that attention).
A key point is that The Attention Economy is about the consumer having choice - they get to choose where their attention is 'spent'. Another key ingredient in the attention game is relevancy. As long as the consumer sees relevant content, he/she is going to stick around - and that creates more opportunities to sell.
Read/WriteWeb has published 4 in-depth articles about the Attention Economy, all of them written by Alex Iskold. We encourage you to read them if you want to learn about the Attention Economy: