The class of distributions that both power law tails and the Normal distribution fall under are called "Levy-stable distributions" [1]. Given the sum of independent, identically distributed (I.I.D.) random variables that converge to a stable distribution, the limiting distribution is Levy-stable (read, power law tails). If you the sum of the I.I.D. random variables has finite variance, the Gaussian distribution is the limiting distribution, which is a special case of the Levy-stable distributions. If each of the I.I.D. random variables has infinite second moment (e.g. infinite variance) and converges, then it converges to something with power law tails.
We've been spoiled to think that the Gaussian's are the only or most common limiting distribution (thus the name "Normal") but Levy-stable distributions are another class of "common" distributions for the same reason: a common limiting distribution of sums of (independent) random variables have a limiting distribution that has power law tails. This was one of Mandelbrot's work horses in that power laws show up in nature, are self similar and represent a kind of "statistical" fractal [2].
There are other ways to generate power law tailed distributions as well, including having a random variable that has an exponentially increasing value with an exponentially decreasing probability of occurring, where each go at a different rate, say [3].
[1] https://en.wikipedia.org/wiki/Stable_distribution
[2] http://www.webofstories.com/play/benoit.mandelbrot/32;jsessi...
Most probability distributions are designed to model some kind of process. The time between clicks of a geiger counter is exponential, the number of clicks in a second is poisson. The amount of time your geiger counter lasts before it breaks and you have to buy a new one is weibull. That's the whole reason why we invented probability distributions in the first place: to model processes.