What does kurtosis measure (not the shape of the distribution).

The distribution of a set of values can be quantified by the skewness (which measures asymmetry) and kurtosis. What does kurtosis measure? It is often said that is quantifies the shape of the peak of the  distribution or the peakedness of the distribution, but Westfall has persuasively demonstrated that this is not true(1).

Understanding how kurtosis is computed can help you understand what it means. These steps compute the kurtosis of a set of values:

1. Subtract the sample mean from each value,  The result will be positive for values greater than the mean, negative for values that are smaller than the mean, and zero for values that exactly equal the mean.
2. Divide each of the differences computed in step 1 by the standard deviation of the values. These ratios (the difference between each value and the mean divided by the standard deviation) are called z ratios. By definition, the average of these values is zero and their standard deviation is 1. 
3. For each value, compute z⁴. In case that doesn't render well, that is z to the fourth power.  All these values are positive. 
4. Average that list of values  by dividing the sum of those values by n-1, where n is the number of values in the sample. Why n-1 rather than n? For the same reason that n-1 is used when computing the standard deviation. 
5. With a Gaussian distribution, you expect that average to equal 3. Therefore, subtract 3 from that average Gaussian data are expected to have a kurtosis of 0. ?This value is sometimes called the excess kurtosis.

Because the z values are taken to the fourth power, only large z values (so only values far from the mean) have a big impact on the kurtosis. If one value has a z value of 1 and another has a z value of 2, the second value will have 16 times more impact on the kurtosis (because 2 to the fourth power is 16). If one value has a z value of 1 and another has a z value of 3 (so is three times further from the mean), the second value will have 81 times more impact on the kurtosis (because 3 to the fourth power is 81). Accordingly, values near the mean (especially those less than one SD from the mean) have very little impact on the kurtosis, while values far from the mean have a huge impact. For this reason, the kurtosis does not quantify peakedness and does not really quantify the shape of the bulk of the distribution. Rather kurtosis quantifies the overall impact of points far from the mean. 

1.    Westfall PH. Kurtosis as Peakedness, 1905–2014. R.I.P. The American Statistician. 2014 Aug 11;68(3):191–5.