Two families, one connection
An address family describes the format of the addresses a connection uses. IPv4 uses the familiar dotted form; IPv6 uses the longer hexadecimal form. Modern operating systems support both at once, which is why a device can have four addresses - a global and a link-local one in each family - and why “do I have IPv6” has more than one answer.
The check above reports two separate things. The first is the family this site saw your connection arrive on, taken from the request itself rather than from anything you typed. The second is whether the browser offered a global IPv6 address as a local peer candidate, which is a statement about the device rather than about the path.
Why the answers can differ from each other
A name that publishes both an A and an AAAA record gives the client a choice, and clients prefer IPv6 where they can. If the IPv6 path is broken somewhere between the device and the destination, the connection attempt fails on that path before the client falls back. The result is a site that loads from one network and not from another, which is the pattern that makes broken IPv6 look like a broken server.
The reverse case is more common than it sounds. A device can hold a global IPv6 address, and the local network can still have no route for it, because the router was given an address without a working prefix delegation. The device then looks IPv6-capable to itself and reaches nothing over IPv6.
A third case is a destination that publishes no AAAA record at all. There is nothing wrong with the client or the path; the destination simply does not offer IPv6, and the connection uses IPv4 by design. Publishers do this deliberately, and an empty AAAA answer is a normal answer, not an error.
Reading the local candidates
Peer candidates are the addresses a browser offers when it is asked to describe how it could be reached. They are produced locally and are not sent to this site. The useful information in them is small but real.
A global IPv6 address in that list means the device has one and the browser is willing to reveal it, which in practice means the network is routing IPv6 rather than merely assigning it. A private IPv4 address means the device is behind translation, which is the ordinary case on home and office networks. Host names instead of addresses mean the browser applied its own privacy substitution; that is a deliberate limitation of this test, not a failure.
Link-local and unique-local IPv6 addresses are filtered out of the list, because they are not reachable beyond the local segment and saying otherwise would be misleading.
What this page does not measure
It does not test reachability from the outside. Whether an address is reachable depends on the router’s firewall and on the provider’s policy, and no query made from inside the network can determine it.
It does not measure throughput or latency on either family. A dual-stack connection can have a much slower IPv6 path, which some operators use deliberately and others discover by accident, and detecting that requires timing requests rather than looking at addresses.
It does not test a different destination. The family reported here applies to this site only; whether a specific service is reachable over IPv6 depends on that service publishing an AAAA record and on your network allowing the path.
What to do with the result
If the connection arrived over IPv6, the family is working for this destination. Whether it works elsewhere is a separate question that this page cannot answer.
If the connection arrived over IPv4 while the device holds a global IPv6 address, the path is the suspect rather than the device. The practical test is to compare a dual-stack name and an IPv4-only name from the same machine; if only the dual-stack one fails, the IPv6 path is the cause and the router’s prefix delegation is the first thing to check.
If the connection arrived over IPv4 and no global IPv6 address was offered, the network is IPv4-only or the browser withheld the candidates. The IP lookup reports the address family the server sees for the same connection, which separates those two cases. For mail servers, whether an address publishes an AAAA record is a separate decision, covered in the DNS record reference; reverse lookups for IPv6 use a different name form, described in the reverse DNS lookup page.