IPv6 (Internet Protocol version 6) is the newest version of the Internet Protocol. Most of the internet we use today uses IPv4, which has the annoying limitation of only allowing 4 billion unique addresses, and we have already run out at the IANA level in 2011 and at the RIR level in the late 2010s.
IPv6 allows for a much bigger, even unlimited number of addresses, however, it is not compatible with IPv4. Your ISP needs to support IPv6, and unfortunately, a lot of ISPs, especially smaller ones, simply do not support IPv6.
Even then, the number of IPv6-enabled websites is relatively small. Sites like Google, Facebook, and even this website are IPv6-enabled, but others, like Reddit and Twitter, are not.
At peterjin.org, we believe that the widespread adoption of IPv6 is essential to the growth of the Internet. Without it, much of the things on my website like IPv6 Things, Traceroute Text Generator, and Socketbox would never exist. These web applications all exploit the large address space of IPv6 to do something that cannot be comparably done in IPv4, and I hope that similar things may start to exist someday, whether it's by me or by someone else.
It is also important to stop thinking of IPv6 as a mere replacement for IPv4 and start thinking about more than just network design. Customers of a product that even use the IP addresses directly don't care about the layout of the network; they just care about what IP address is needed to access a particular service. For this reason, I often conjecture that "IP addresses should define services, not hosts." This is true for both IPv4 and IPv6.