"Late in the afternoon, a few days after New Year, Harry, Ron, and Ginny lined up beside the kitchen fire to return to Hogwarts. The Ministry had arranged this one-off connection to the Floo Network to return students quickly and safely to the school." (HBP17)
A division of the Department of Magical Transportation which regulates, controls and appears to maintain the Floo Network
The Floo Network Authority is located on Level Six of the Ministry of Magic headquarters.
It maintains and regulates the Floo Network. Employees of the Floo Network Authority have the ability to monitor Floo connections, and can eavesdrop on Floo conversations in a manner analogous to wiretapping a Muggle telephone. Witches and wizards working for the Floo Network Authority are referred to as Floo Network Regulators (OP28).
It is against regulations for a Muggle fireplace to be connected to the Floo Network (GF4), probably due to the likelihood of violations of the International Confederation of Wizards' Statute of Secrecy in the event of a connection becoming active.
Travelling by Floo Powder through the Network means that wizards can travel from place to place without breaking the International Statute of Secrecy, without the danger of serious injury (unlike Apparition) and is suitable for children, old people and the ill (Pm).
The Floo Regulation Panel (GF4) controls which fireplaces are connected to the Floo Network. This apparently cannot be done by even a qualified wizard without privileged access and/or special magical tools, because Sirius Black had to break into a wizarding house to use the Network rather than simply starting a fire in the cave where he was hiding at the time and using that (GF19). Connecting a fireplace to the Network does not require that the Floo Network Authority visit that fireplace, and can be done temporarily for as short a period as a single afternoon, as was once done with the fireplace at number four, Privet Drive (GF4) and with fireplaces at Hogwarts during Harry's sixth year (HBP17). Ordinarily such a temporary connection may be difficult to arrange; Arthur Weasley had "a useful contact at the Floo Regulation Panel" arrange the temporary connection for number four, while the Hogwarts connection was part of a security measure to protect students returning to school after the Christmas holidays.
Marietta Edgecombe's mother was in charge of monitoring the Hogwart's fires for unauthorised connections to the Floo Network (OP27) while Dolores Umbridge was Defence against Dark Arts teacher at Hogwarts.
In 1855, wizards were afraid to use the Floo Network due to the mysterious disappearance of Violet Tillyman. However, the problem turned out to be due to not speaking the name of her destination clearly enough and then having a good reason for not letting her location be known (Pm). Harry Potter ended up in Knockturn Alley after a mishap with Floo Powder (CS4).
A "flue" is a pipe or duct running up a chimney from a fireplace or from a boiler. It vents the hot air or gases outside.
It is not clear how the regulations define a "Muggle household"; in this case, number four was considered a Muggle household although a wizard was then in residence. The regulation may be that at least one legally adult witch or wizard must be in residence in a household for it to qualify as a wizarding household, and/or that he or she must be the property owner/ leaseholder, but this is speculation rather than fact. It is somewhat disquieting that a connection to number four could be arranged without the knowledge or consent of the people living there, and without (as far as we know) any enchantments being required at the Privet Drive fireplace to establish the connection.