For example the distribution of weights of human. There are not many adults under 40 kg, but a lot more people heavier than 100 kg, although the average of an adult's weight is, let's say, 70 kg. Another example is this human reaction time, sharing the same tail property. I can image a "mixture" of normal distribution and Cauchy distribution, where the normal distribution dominates at the left side and the Cauchy distribution dominates at the right side, but is there already a family of such distributions with good properties, like, that we can easily calculate the mean, variance, entropy, etc?
There are infinitely many distribution families with heavier right tails and lighter left tails.
A few are well known. I'll mention a few, but don't by any means think this would be all. Some of these have indeed been used for things like reaction-times
Distribution Simple form for Family Mean Var Entropy Gamma Yes Yes Yes Lognormal Yes Yes Yes Chi Yes Yes Yes (not χ², but its square root) Fréchet Yes Yes Yes Weibull Yes Yes Yes (not all members are right skew) Gumbel Yes Yes Yes log-logistic Yes Yes ? F Yes Yes ? (Fisher-Snedecor F) Skew Normal Yes Yes ? ExGaussian Yes Yes ?
Here "simple form" means "is available in a form that may be readily calculated with common software", so things like Gamma functions count as 'simple'. In the above table "?" means "I didn't quickly locate one" (which suggests that simple forms may either not exist or be hard to calculate)
See the list of distributions at Wikipedia here -- the individual pages generally list the mean and variance and sometimes give the entropy.