Just over a year ago IPv6 was turned on for a day just to check that everything worked. Today, June 6th 2012, IPv6 has been switched on for good. This is the day the Internet got big enough for everyone - with 340 billion, billion, billion, billion addresses.
IPv6 takes over from IPv4 today in the sense that it is the new standard way of using the Internet and in time we will all make the change. Back in February 2011 the IANA ran out of old IPv4 32-bit addresses and any that you get now are reassignments of addresses already allocated but not used.
You can appreciate that we need more addresses than 32 bits can provide but perhaps moving to 128-bit addresses is over provision. What the huge address space is suppose to be used for isn't people but machines. IPv6 makes IP addresses so abundant that we can afford to give any machines that need to be connected their own unique address - so making the Internet of Things possible.
With luck you shouldn't have noticed any difference after 00:01 GMT on June 6, 2012 when the switchover happened. In most cases you don't even have to do anything to make it continue working, but you will have to change things eventually. At the moment you can access IPv6 resources using address translation, but sooner or later your ISP will upgrade to IPv6 and so will hosting services.
Slowly the Internet will make the transition to the new protocol. Until it does then the real benefits of IPv6 won't be apparent because the interoperation is achieved by sharing existing IPv4 addresses and other inefficient tricks.
You can check if your connection has IPv6 by visiting a testing site provided by Google: Test My Connection.
Vint Cerf, Chief Internet Evangelist at Google and one of the inventors of the original IPv4-based Internet, summed it up very nicely on the Google Blog:
"Today we launch the 21st century Internet: you ain’t seen nothin’ yet."
If you would like to see Vint Cerf explain why he thought 32-bit addressing would be enough, and why he changed his mind, watch this video: