"Heavy tailed" is a bit tricky in the sense that there is no unique definition of what it really means. Usually, it refers to the MGF being infinite on the side of zero that the heavy tail is (i.e. right tail, MGF infinite for arguments $> 0$ etc.). However, as is pointed out by Mikosch in one of his papers (I forget which one), the notion of heavy-tailedness depends on the model under consideration.

Commonly used versions of heavy-tailed random variables are those which are $(i)$ regularly varying and $(ii)$ subexponential ($(i)$ being a subclass of $(ii)$). Regularly varying refers to the tail of the distribution being regularly varying in the usual sense from analysis. The perhaps easiest example of such a distribution is the Pareto distribution. That a distribution is subexponential means that

$ \lim _{x \to \infty} \frac{P (S_n > x)}{n P (X_1 > x)} = 1,$

where $S_n$ is the sum of $n$ i.i.d. random variables and $n=2,3,...$ Example of such a distribution is the Weibull distribution.

Having heavy tails basically means, as the name hints, that the tail of the distribution decays slower "than normal". E.g., in the case of a regularly varying distribution the decay is according to a power law.