Right now, most of the Internet uses something called IPv4. IPv4 is an old standard. It has some design flaws, including not enough IP addresses to adequately cover current demand.
IPv6 is intended to replace IPv4. It solves a lot of the design problems with IPv4 and it is highly unlikely it will ever run out of available unique IP addresses.
I have known about IPv6 for over a decade, but I just recently started migrating to it. My web servers are now all running IPv6 (as well as IPv4) and my home network is ready to go IPv6 but I need to replace my router.
Non technical people probably should wait until their Internet Service Provider starts offering IPv6 but for the technically inclined, if you are not yet IPv6 ready, have a look at http://ipv6.he.net/. They have a FREE program to help you get up and running both from home (even if your ISP does not yet offer it) and on your web servers. And if you complete the program (reach level of sage) you get a free T-Shirt!
I started the program yesterday, and got to Sage in a single day:
If you need an IPv6 ready host, check out Linode - Excellent customer service, and most of their data centers will assign you both an IPv4 and IPv6 address for your account.