To make computers and other devices communicate with each other over the Internet, they must all use the same set of rules, the same protocol. The protocol used is called IP. Today's Internet mainly uses version 4 of the Protocol, simply called IPv4.
Because the number of addresses in IPv4 is limited, the future Internet will use the next-generation IP, IPv6. (IPv5 is also available, but is not used.) With IPv6 the IP addresses are 128 bits long instead of 32, meaning that the total number of possible addresses is almost unlimited.
To demonstrate this, you could simply explain it this way. With IPv6, there could be about 5 x 10 to the power of 28 IP addresses for every person on earth. Then you have your very own 50 0000 0000 0000 0000 0000 0000 0000 0000 0000 IP addresses!
The Bredbandskollen IPv6 measurement works the same way as the regular one and can be accessed via the IPv6 button on the top menu. If you are returned to this page after clicking the button, then you probably do not have access to IPv6.
(If you are absolutely sure that you have IPv6 but still get this message, try using this link: ipv6only.bredbandskollen.se)