For those of you unfamiliar with the issue and how the internet works (in simple terms) each and every connected device on the internet needs an IP address. It's like a phone number, basically a unique number given to your computer, mobile phone, or whatever, whenever it is connected to the internet. It then "dials" another number whenever you check your email, go to a web page or any other activity.
Of course, to dial a number you need a number your self. And as we all know the number of connected devices on the internet is by no means slowing down. So we need more numbers to allocate to these devices. But here's the catch: There are no more numbers available. There are some, but not many. So that implies that the internet is full. All thought there are many unused numbers held by organizations that ones got them but are no longer using them, these unused numbers are not enough for any sustainable future growth.
The current numbering system we're using is called IPv4. This system contains about 4,000,000,000 unique available addresses in total. At the time of writing, about 2% is unallocated. At the speed things are going, that will be empty within a matter of days. Luckily this development hasn't come as a shock to the people who help maintain the internet, so there's a new version available called IPv6.
IPv6 contains 3.4×10^38 addresses. Compared to IPv4's 4.3×10^9 that's a lot more. So much more that it's deemed enough for the foreseeable future. So why isn't this similar to just dialing a number or more on your phone when you make a phone call? It basically comes down to upgrading all the connected equipment. That includes your computer, your phone, your home router, all the network equipment your ISP have installed, etc. And since all the ISP's need to talk to each other they need to upgrade the systems used between them, and so on.
Luckily, as IPv6 has been available for quite some time, you computer probably already know how to "speak" IPv6. Your home router might also know how to do this, but chances are you might need a new one. This is probably provided by your ISP so go talk to them about it. As I'm quite sure they are still interested in selling you access to the internet they should have some migration plan in place.
But what if you're stuck on some lame ISP with incompetent and slow moving people? Do you really have to wait for them to get going? Ideally, yes, but.. It's possible to tunnel IPv6 over IPv4. It's like dialing a 8 digit number first, and then, when the other end picks up the phone, you dial a 12 digit number in addition. Wikipedia maintains a list of IPv6 tunnel brokers, so if this lack of IP addresses is starting to bother you, or you just want it now, go there and check it out.
In his opening remarks to the conference, Google's Chief Internet Evangelist Vint Cerf urges ISPs to move to IPv6, so that a "black market" for Internet addresses won't occur.