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:
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This makes sense. Robert O. Briggs has been writing about limited attention resources since the mid 1990s. The next economy (after capital, service, and information economies) is going to be an attention economy.
In short, information is no longer a scarce resource. We have too much of it - and we keep getting more. Therefore, we are no longer in an information economy. What provide value are tools that make better use of our limited ability to pay attention to all the stimuli around us.
By the way, with different terminology, this is what Web3.0 is all about.
I think that subsequent to paying attention, getting consumers to make 'gestures' in reaction to or in relation to the information is going to be increasingly important as well. It's one thing to filter down all the information, it's another thing to get people to interact with it.