You used to be able to buy blocks of public addresses from ARIN (and I guess you still can[0] but only under special circumstances[1]) but the IPv4 public assignment table is essentially full. You could buy blocks in class A (i.e you would own X.<all>.<all>.<all>), class B (X.Y.<all>.<all>), or class C (X.Y.Z.<all>).

The class assignments themselves are now either a result of an RFP (for special class IP blocks like multicast) or historical significance (basically, who grabbed it when it was available). XKCD had a comic in 2006 mapping the IPv4 assignment table[2].

I don't know much about IPv6 other than that its address space is so large that you essentially don't need to use NAT anymore, so I doubt we'll ever deplete it.

0. https://www.arin.net/fees/fee_schedule.html

1. https://www.arin.net/resources/request/ipv4_countdown.html

2. https://xkcd.com/195/