a normalized fourth moment of a distribution or dataset.

Kurtosis refers to the fat-tailed-ness of a distribution. It is often defined as a normalized fourth central moment $\mu_4$ of a distribution or dataset. It can be any non-negative real number or even (for distributions) infinite.

There are several flavors of kurtosis commonly encountered, including the kurtosis proper, denoted $\beta_2$ (Abramowitz and Stegun 1972, p. 928) or $\alpha_4$ (Kenney and Keeping 1951, p. 27; Kenney and Keeping 1961, pp. 99-102) and defined by:

$$\beta_2 = \frac{\mu_4}{\mu_2^2}$$

where $\mu_i$ denotes the $i$th central moment (and in particular, $\mu_2$ is the variance).

Note that kurtosis does not measure the "peakedness" of a distribution (Westfall, 2014), as is commonly believed.

Sometimes "kurtosis" refers to the excess kurtosis, defined as $\beta_2 - 3$. This is the amount by which the kurtosis differs from that of any Normal distribution.

Reference: mathworld.wolfram.com

Excerpt reference: statistics.about.com

Westfall, P. H. (2014). Kurtosis as Peakedness, 1905–2014. R.I.P. The American Statistician, 68(3):191-195, DOI: 10.1080/00031305.2014.917055