The (nearly) 256^4 = 4+ billion IPv4 addresses identifying the computers on the Internet are gone and this system of identification has therefore entered the cap-and-trade system. ;-)
Microsoft has already bought some allowances. One IP address is about 11 dollars, much like a ton of CO2 emission rights. ;-)
Many ad hoc tricks exist to circumvent the capacity limitation but a new global system that will simply eliminate the need for tricks may eventually become preferred. The IPv6 system - designed to distinguish 2^128 hosts - not far from 10^123 bits that can be stored in the visible patch of our (nearly) de Sitter space - is the leading candidate. Operating systems and, to a lesser extent, routers have been trained to be compatible with it for a decade.
However, only a few percent of the Internet traffic is running through IPv6 at this moment - it is mostly peer-to-peer exchange of the data and most of it is arguably violating someone's copyrights. Turning IPv4 off would almost surely mean an unbelievable hassle at this moment. Fortunately, it's not needed.
On Wednesday, the world will celebrate the IPv6 day and you will be allowed to find out how huge a debacle it would be for you if the IPv4 addresses were suddenly turned off.
Don't be afraid. IPv4 won't be turned off tomorrow. But various providers and Internet companies will try to be available via IPv6. You may try ipv6.google.com. If you can open it, you're just doing great. I can't. ;-) Internet Explorer tends to use some bridge or whatever it is, teredo, but I think it's not too useful.
You may only have problems tomorrow if your computer decides that IPv6 is cool and working and decides to switch to it - while another part of the network is not ready. In the case you will have problems with your connection, find your connection settings in a control panel and turn IPv6 off.
Are there some regions that are getting ready for IPv6? Well, there is one. CZ.NIC has just announced that *.cz is the most technologically advanced top-level domain of the Internet when it comes to the percentage of the content that is already available via IPv6: it's eight percent! The percentage has quintupled during the last year - it's almost like the solar panels! :-)
To compare, *.de, *.com, and *.net domains only have one percent of the content available via IPv6 these days. You may try to test your connection at test-ipv6.com, get a more optimistic message at ipv6test.google.com (my connection passes even without an IPv6 address) or read some other tips from Google and 10 tricks from PC World that are related to the IPv6 day.