IPv6 is the successor to IPv4, the Internet addressing protocol which has been used for many years since the early days of the Internet. When the Internet was first founded, it was established as a research network, and the addressing was limited. It was never thought that it would be used to connect everything from a mobile phone to a hi-fi and fridge to the 'net. As such, it is estimated that we will run out of the existing IPv4 based addresses in 2012, so a new protocol has been developed that will work around this problem. It also introduces some new features to improve how the Internet works.
An IP (Internet Protocol) address is used on the Internet to contact a computer in a similar way that a telephone number is used to connect to a telephone line. Each IP address is unique and any data that is sent to that address will be routed through the Internet to arrive at the destination of that computer. Most people will be more familiar with easy to remember hostnames such as www.google.com which are looked up in a directory called DNS (similar to a yellow pages) and translated in to the IP address. You can often use the IP address instead of the hostname (for example typing http://80.249.99.130/ (or http://[2a02:68:1::4]/ for IPv6 users) in you address bar will bring you to this website.
The current IPv4 address space contains 4.3 billion addresses. The number of addresses offered by IPv6 is 340,282,366,920,938,000,000,000,000,000 billion (2 to the power of 128) which means that the size of the Internet could double every year and we would still have enough addresses for the next 96 years.
There are two ways to use IPv6, natively and tunnelled. Natively means you connect directly over an IPv6 network and would require your service provider to support it (few currently do). To connect natively you will also need a broadband router to support IPv6, and there are currently few of these available. Tunnelled IPv6 is a way of tunnelling IPv6 over an IPv4 network so you can connect to IPv6 services. Most operating systems currently support both and Windows Vista and Windows 7 both allow you to connect over a Teredo tunnel.
IPv4 exhaustion is happening. Once there are no more IP's left, providers will either need to start providing IPv6 addresses to users or will resort to using NAT. NAT doesn't give users a real IP address and can break certain applications like online gaming or VoIP. Adopting IPv6 will make sure you are ready for it. There is no doubt that IPv4 will be around for a long time, but once IPv6 grows, those not able to use IPv6 may find that parts of the Internet are inaccessible.