What do heavy-tailed distributions look like and how do they relate to the central limit theorem? Which modes of convergence imply which, with counterexamples?

What do heavy-tailed distributions look like, for example power laws and Levy stable laws with index alpha below 2, and how do they relate to the central limit theorem? Then state which modes of convergence of random variables imply which, and give counterexamples for the implications that fail.