The other kind of society is one in which the best way to establish a secure place for yourself is to take secure control of some piece of income generating property. Piketty uses Jane Austen's novels to show what these societies look like: marriage and other political arrangements become far more important than how hard working you are. Hard work and toil is generally associated with the poor.
So Piketty's argument is that, while we may believe we are in a pull-yourself-up-by-your-bootstraps society of hard work, the numbers show that we are in fact in a work-your-political-connections-to-establish-control-over-productive-capital society.
So it should be no surprise that HN is a great place to find counterarguments for Piketty's theories. :) But that's his theory, and I believe it's worth keeping an open mind as to whether it's true. If it were, we would see that it's easier to get richer by already being rich or making friends with the rich than it is to get rich by working to build something yourself.