Today, 6 06 2012 is World IPv6 Launch day, the day that several large websites, ISPs and CDNs turn on support for IPv6 by default.
What this means is that devices properly configured to use IPv6, running on IPv6 networks, will be able to connect to websites via IPv6 addresses rather than old IPv4 ones without any special intervention.
Everything will work out of the box, but, again, there are plenty of caveats. Assuming your operating system and your router handle IPv6, as they should in this day and age, all you need is an ISP that offers services via IPv6.
There aren't many of those around and even the ones that do support it do it in an optional or experimental way.
After World IPv6 Launch
day, any ISP involved will offer IPv6 services to at least some of their customers. If you're one of the lucky ones, you'll be able to visit Google, YouTube, Facebook and several other large websites over an end-to-end IPv6 connection.
The need to switch to IPv6 is tied to the dwindling number of available IPv4 addresses. IPv4 was designed three decades ago and only allows for 4.3 billion (2^32) devices connected to the same network.
So, if the network is the internet, the number isn't that high so we've pretty much run out. ISPs have to resort to ugly hacks (NAT) to get everyone online.
With IPv6, each device connected to the internet can get its own address, whether this means your laptop, your phone or your thermostat. With every device accessible to every other device on the network, a number of new interesting services can finally develop.
But it's not just that it has more addresses, which it does, IPv6 can assign 2^128 addresses, that's some 340 trillion, trillion, trillion addresses. Needless to say, that's going to be enough for anything we can come up with up until we discover aliens and have to connect to the, well, galaxynet or whatever it is going to be called.