They are influential people:
“Look, Albus,” said Fudge, uncomfortably. “Hagrid’s record’s against
him. Ministry’s got to do something — the school governors have been
in touch —” “Yet again, Cornelius, I tell you that taking Hagrid
away will not help in the slightest,” said Dumbledore. ~Harry Potter and the Chamber of Secrets - Cornelius Fudge
In a situation where muggleborn students are petrified at Hogwarts and there are rumors about the Chamber of Secrets being reopened Fudge is concerned mainly about the impression he would make on the governors. This implies they are the ones with power that could affect his political career.
They are dominated by Lucius Malfoy
It seems Lucius Malfoy is capable of coercing even those governors whom he can't convince or bribe:
Very strange tales they told me, too. . . . Several of them seemed
to think that you had threatened to curse their families if they
didn’t agree to suspend me in the first place.”~Harry Potter and the Chamber of Secrets - Dobby's Reward
Which probably means there aren't any Order members among the governors or people who are loyal to Dumbledore and trust him, else they would've simply asked his protection, instead of submitting to Lucius Malfoy's threats. Nor we ever hear about any gryffindor student having a parent in the board. (In fact the only parent we hear about being a governor is Draco's father).
They only decided on reinstalling Dumbledore and going against Lucius when a known ministry official's pureblood daughter was attacked:
" “So!” he said “You’ve come back. The governors suspended you, but
you still saw fit to return to Hogwarts.” “Well, you see, Lucius,”
said Dumbledore, smiling serenely, “the other eleven governors
contacted me today. It was something like being caught in a hailstorm
of owls, to tell the truth. They’d heard that Arthur Weasley’s
daughter had been killed and wanted me back here at once. They
seemed to think I was the best man for the job after all. ~Harry Potter and the Chamber of Secrets - Dobby's Reward
It's pure speculation, but it seems the governors are chosen from the more conservative, influental and wealthy members of the wizarding society, those people who probably symphatize somewhat with Lucius' views.
After all, in the book Chamber of Secrets Arthur Weasley’s Muggle Protection Act is introduced. The more conservative pureblood families represented by Lucius Malfoy are opposed to this proposal and Lucius uses Riddle's diary and his influence on the governors board to harm his muggle-loving political opponents: Arthur Weasley and Albus Dumbledore. But then things go too far, Arthur Weasley’s daughter is attacked and the governors decide that enough is enough.