FQDN is an acronym for Fully Qualified Domain Name. You can also refer to these as sub-domains or third level domain names. The important distinction is that an FQDN is unambiguous. Let's use the FQDN 'backup.2xdata.net' as an example. It clearly identifies the hostname (backup) and domain (2xdata.net) of a server and allows us to resolve this to an IP address using DNS:
# dig backup.2xdata.net
;; QUESTION SECTION:
;backup.2xdata.net. IN A
;; ANSWER SECTION:
backup.2xdata.net. 300 IN A 184.108.40.206
As you can see in the answer section there is no ambiguity: 'backup.2xdata.net' is a DNS A record that resolves to the IP address '220.127.116.11'.
So the short answer is that an FQDN is a DNS name that points to an IP address. You can set up a sub-domain of any domain you own by contacting whoever hosts your DNS. In most cases this is also the registrar, but it might also be your hosting provider (whoever hosts the website that the domain points to), or in some cases a 3rd party DNS service like DynDNS. In most all cases these companies have online control panels to facilitate these updates.
Why do we prefer to use an FQDN?