Part of the reason for this excitement is because there are many bad decisions being made in the world today, and as the world gets more populated, a deep scientific understanding of why human beings often make such "predictably irrational" decisions will help influence public policy. The field of Heuristics is a kissing cousin of neuroeconomics but does not involve the rigorous analysis of cognitive behavior and the biochemical processes taking place in the human brain when making a decision.
Neuroeconomics combines psychology, economics, and neuroscience, to study how people make decisions. It looks at the role of the brain when we evaluate decisions, categorize risks and rewards, and interact with each other. It can be included in the field of social neuroscience.
Behavioral economics experiments record the subject's decision over various design parameters and use the data to generate formal models that predict performance. Neuroeconomics extends this approach by adding observation of the nervous system to the set of explanatory variables.
In a typical behavioral economics experiment, a subject is asked to make a series of economic decisions. For example, a subject may be asked whether they prefer to have 45 cents or a gamble with a 50% chance of one dollar and 50% chance of nothing. The experimenter will then measure different variables in order to determine what is going on in the subject's brain as they make the decision.
Neuroeconomics has developed into an up and coming field in graduate studies. Several universities are conducting direct research on the field, such as New York University, Duke University, and George Mason University. Furthermore, some programs actually offer a degree in Neuroeconomics. Claremont Graduate University is the first institution to offer a PhD in Neuroeconomics; it is ranked as one of the best Neuroeconomics institutes in the United States. Caltech now (c 2007) has a Behavioral and Social Neuroscience (BSN) PhD in either CNS or HSS, mixing economic theory, neurobiology, computational neuroscience,dynamic causal modeling and neuroscientific techniques.
If you would like to read more about Neuroeconomics, a good reading primer is a book called Your Money and Your Brain: How the New Science of Neuroeconomics Can Help Make You Rich, by Jason Zweig.
More detailed information on Neuroeconomics can be found on Wikipedia.