You'll know that most of the Internet runs on IP version 4, the first cut of the Internet Protocol to see widespread use. It was standardised in 1981; for the past twenty five years or so, it's underpinned the beyond-massive expansion of the Internet.
You'll also know, if you've heard of IPv4, that a long time ago it was realised that it was going to hit problems at some point. The headline issue is address space: with its 32-bit field, there can be something over four billion IP addresses on the same network. At the time it was designed, that seemed like enough for everything you could ever want to do on a network. Now, even with various ways to reuse common addresses on subnets, it's grossly inadequate. There are, after all, something like three billion mobile phones alone in use in the world.
So in 1996, a new version IPv6, was created. This has enough addresses for the entire universe; the idea was that as it got adopted, first in the backbones and then out to the edges of the network, enough IP nodes would move across to ease the address crunch and, in time, allow IPv4 to quietly fade away, like black and white television in the face of colour.
Hasn't happened. Now, the consensus is that IPv4 will continue to be important for ten to twenty years at least - possibly for the foreseeable future - while the address space will run out in two to four years. As far as can be ascertained, IPv6 won't be in any position to solve that problem: if anything, it will add to it - we'll have to run the two systems side by side, with all manner of unforeseen management and security problems arising from their interaction.
It gets worse. The developed world has the lion's share of IPv4 addresses, while the developing world has the greatest demand. There are ideas for managing the address space more efficiently by introducing auction and other pricing mechanisms to encourage better use (people who don't need their allocation will flog them off rather than hoarding them, while new uses will be parsimonious in their approach), but the developing world sees this as unfair in the extreme. You can see their point.
There are other problems: how do you route IP addresses when the existing hierarchy breaks down due to address spaces moving through the network? Who's responsible for managing an increasingly incoherent network? Who foots the bill when your address space is sold from underneath you? In any case, it doesn't solve the basic problem - it merely makes it increasingly expensive to innovate.
Many of the same issues occur if you decide just to manage the existing resources more carefully, by using more address translation and other tricks, what's left can last longer but at the expense of manageability and interconnection. It won't help arbitrate between conflicting interests, and the expense and inconvenience will increase until things just stop working.
The third option, to aggressively move as much as possible to IPv6 as quickly as possible, involves enormous up-front costs and requires tons of research to work out how to do it. It doesn't address the fast-accelerating user base of existing IPv4 nodes, many of which will be difficult or impossible to safely upgrade, nor the fact that the greatest expense will be born by new networks - another barrier to innovation and the spread of the network into the poorer parts of the world. It is, however, the best chance of a permanent solution.
In short, we're heading towards a classic resource depletion, with the haves at loggerheads with the have-nots. When IPv4 addresses do run out, the implications are huge: much commercial development will stop and many of the aspects of the Internet we've taken for granted will change. This will happen by 2010 - at best, by 2012 - and to form a consensus on what to do when that happens will take an enormous effort that would normally take tons of time. Which, in this case, we just don't have.
(You could almost see it as a dry run for global warming and peak oil, if you were sufficiently apocalyptic.)
For more on this, take a look at this PDF from the Japanese government, which looks at all this in some detail. It makes engaging reading.
(thanks to Izumi Aizu of the Institute for HyperNetwork Society, Tama University, Tokyo, for sending that out on David Farber's Interesting People mailing list)
[UPDATE: If you found that interesting, check out Randy Bush's impassioned, clear and even more engaging presentation on just how badly IPv6 has not helped matters, He compares it to the Iraq invasion - no planning, no costing, no support for the people on the front line, and in need of radical reform as quickly as humanly possible, Phew. And who is Randy Bush? Old school nethead of the finest kind. Pay attention.]