A group of twelve wizards charged with overseeing the running of Hogwarts. They have the power to suspend or fire Headmasters (CS14), and can decide to shut down Hogwarts if the school is deemed unsafe – a situation that arises when the Chamber of Secrets is opened (both times), when students keep getting hurt by Draco’s attempts on Dumbledore’s life (HBP19), and when Severus Snape kills Albus Dumbledore (HBP29).
They also appear to have some authority regarding staff appointments, given Hagrid’s worries about them not liking Buckbeak injuring Draco (PA11). But they certainly don’t have absolute veto powers over the hiring of teachers, given that Dumbledore’s unconventional choices always went seemingly unchallenged.
The only known member is Lucius Malfoy, who was sacked in 1993 after coercing the other governors into getting rid of Albus Dumbledore (CS18).
The power structure is very unclear among the Board of Governors, the Ministry of Magic, and the Headmaster of Hogwarts. Unfortunately, the Board of Governors is completely absent in Order of the Phoenix, which is where the power struggle comes to a head.
The Headmaster seems to be beholden to the Board of Governors, whose authority can be superseded by the Ministry of Magic. When the Board of Governors fires Dumbledore (CS14), Fudge seems unable to override them. However, he seems to wield authority over Hogwarts just fine in Order without the governors. Most likely, the governors' authority being overridden has to be codified into law as Educational Decrees, which is why Fudge couldn't do anything in Chamber of Secrets.