When I think of 'fat tails' or 'heavy tails' I typically think of situations that can be described by probability distributions with heavy mass in the tails. This might imply that the tails are 'fat' or 'thicker' than other situations with less mass in the tails. (for instance, a normal distribution might be said to have thin tails while a distribution with more mass in the tails than the normal might be considered a 'fat tailed' distribution.)
Basically when I think of tails I think of 'extreme events', some occurrence with an excessive departure from what's expected on average. So, if there is more mass in a the tail of a probability distribution (it is a fat or thick tailed distribution) that implies that extreme events will occur with greater probability. So in application, if I am trying to model or assess the probability of some extreme event (like a huge loss on an investment) then I better get the distribution correct in terms of tail thickness.
See here for a very nice explanation of fat tailed events from the Models and Agents blog: http://modelsagents.blogspot.com/2008/02/hit-by-fat-tail.html
Also here is a great podcast from the CME group related to tail hedging strategies: (May 29 2014) http://www.cmegroup.com/podcasts/#investorsAndConsultants
And here's a twist, what if I am trying to model the occurrence of multiple events simultaneously? (a huge loss in commodities and equities simultaneously, or losses on real estate in Colorado and Florida simultaneously). I would want to model a multivariate process that captures the correlation or dependence between multiple extreme events (in other words between the tails of their distributions). Copulas offer an approach to modeling tail dependence, and again, getting the distributions correct matters.
When I think of how do we assess or measure tail thickness in a data set, I think of kurtosis. Rick Wicklin recently had a nice piece discussing the interpretation of kurtosis and relating kurtosis to tail thickness.
"A data distribution with negative kurtosis is often broader, flatter, and has thinner tails than the normal distribution."
" A data distribution with positive kurtosis is often narrower at its peak and has fatter tails than the normal distribution."
The connection between kurtosis can be tricky and kurtosis cannot be interpreted this way universally in all situations. Rick gives some good examples if you want to read more. But this definition of kurtosis from his article seems keep us honest:
"kurtosis can be defined as "the location- and scale-free movement of probability mass from the shoulders of a distribution into its center and tails. " with the caveat - "the peaks and tails of a distribution contribute to the value of the kurtosis, but so do other features."
But Rick also had an earlier post on fat and long tailed distributions where he puts all of this into perspective in terms of the connection to modeling extreme events as well as a more rigorous discussion and definition of tails and what 'fat' or 'heavy' tailed means:
"Probability distribution functions that decay faster than an exponential are called thin-tailed distributions. The canonical example of a thin-tailed distribution is the normal distribution, whose PDF decreases like exp(-x2/2) for large values of |x|. A thin-tailed distribution does not have much mass in the tail, so it serves as a model for situations in which extreme events are unlikely to occur.
Probability distribution functions that decay slower than an exponential are called heavy-tailed distributions. The canonical example of a heavy-tailed distribution is the t distribution. The tails of many heavy-tailed distributions follow a power law (like |x|–α) for large values of |x|. A heavy-tailed distribution has substantial mass in the tail, so it serves as a model for situations in which extreme events occur somewhat frequently."
Also, somewhat related, Nassim Taleb, in his paper on the precautionary principle and GMOs (genetically modified organisms) discusses such concepts as ruin, harm, fat tails and fragility, tail sensitivity to uncertaintly etc. He uses very rigorous definitions of these terms and determines that there are certain things like GMOs that would require a non-naive application of the precautionary principle while other things like nuclear energy would not. (also tune into his discussion of this with Russ Roberts on Econtalk- more discussion on this actual application at my applied economics blog Economic Sense).