What is IPv6?
The short answer to that is IPv6 is the future.
The longer answer is that it's the replacement for the current scheme for addressing machines in the Internet (that's rather an inaccurate description, since it's more than just that, but it will do for now).
The current scheme, known as IPv4 (version 4 of the Internet Protocol) provides 232 IP addresses (about 4.3 billion). According to the statistics from Internet World Stats there were nearly 2.5 billion internet users in 2012. Given that many people have more than one Internet connected device (computer, phone, tablet computer, printer) those 4.3 billion addresses are rapidly being used up. That's without considering all of the servers and routers that keep the Internet running (Google alone was estimated to have nearly one million servers in 2012).
So, with that (and several other ideas) in mind, a new system was devised, IPv6 (there was no IPv5, in case you're wondering). It solves many problems that exist in the Internet as we currently know it (and probably causes a whole pile of new ones), but the big one is the change in the number of available addresses. There are a lot, lot more of them (2128). If you want help visualising that, Aaron Toponce's The Sheer Size of IPV6 page does it very well.