Heavy tail means that there is a larger probability of getting very large values. So heavy tail distributions typically represent wild as opposed to mild randomness. An increasing variety of outcomes is being identified to have heavy tail distributions, including income distributions, financial returns, insurance payouts, reference links on the web, etc. A particular subclass of heavy tail distributions is power-laws, which means that the PDF is a power. A technical difficulty is that not all moments exist for these distributions, which typically means that quantiles and other order statistics are used instead. It also means that the central limit theorem no longer holds. In its place is a new standard limit distribution for linear combinations such as means, namely the stable distribution.
Levy alpha-stable distribution
Moments of all orders exist for a long tail distribution, but grow very quickly.