One of my favorite descriptions comes from Beej's guide to network programming, something I first read probably in the early to mid 2000s. https://beej.us/guide/bgnet/html/#ip-addresses-versions-4-and-6

3.1 IP Addresses, versions 4 and 6 In the good old days back when Ben Kenobi was still called Obi Wan Kenobi, there was a wonderful network routing system called The Internet Protocol Version 4, also called IPv4. It had addresses made up of four bytes (A.K.A. four “octets”), and was commonly written in “dots and numbers” form, like so: 192.0.2.111.

You’ve probably seen it around.

In fact, as of this writing, virtually every site on the Internet uses IPv4.

Everyone, including Obi Wan, was happy. Things were great, until some naysayer by the name of Vint Cerf warned everyone that we were about to run out of IPv4 addresses!

(Besides warning everyone of the Coming IPv4 Apocalypse Of Doom And Gloom, Vint Cerf14 is also well-known for being The Father Of The Internet. So I really am in no position to second-guess his judgment.)

Run out of addresses? How could this be? I mean, there are like billions of IP addresses in a 32-bit IPv4 address. Do we really have billions of computers out there?

Yes.

Also, in the beginning, when there were only a few computers and everyone thought a billion was an impossibly large number, some big organizations were generously allocated millions of IP addresses for their own use. (Such as Xerox, MIT, Ford, HP, IBM, GE, AT&T, and some little company called Apple, to name a few.)

In fact, if it weren’t for several stopgap measures, we would have run out a long time ago.

But now we’re living in an era where we’re talking about every human having an IP address, every computer, every calculator, every phone, every parking meter, and (why not) every puppy dog, as well.

And so, IPv6 was born. Since Vint Cerf is probably immortal (even if his physical form should pass on, heaven forbid, he is probably already existing as some kind of hyper-intelligent ELIZA15 program out in the depths of the Internet2), no one wants to have to hear him say again “I told you so” if we don’t have enough addresses in the next version of the Internet Protocol.

What does this suggest to you?

That we need a lot more addresses. That we need not just twice as many addresses, not a billion times as many, not a thousand trillion times as many, but 79 MILLION BILLION TRILLION times as many possible addresses! That’ll show ’em!

You’re saying, “Beej, is that true? I have every reason to disbelieve large numbers.” Well, the difference between 32 bits and 128 bits might not sound like a lot; it’s only 96 more bits, right? But remember, we’re talking powers here: 32 bits represents some 4 billion numbers (232), while 128 bits represents about 340 trillion trillion trillion numbers (for real, 2128). That’s like a million IPv4 Internets for every single star in the Universe.