The Order of the Basilisk was based in Voldemort's Secret Hideout.
Voldemort named it after the late Basilisk that lived in the Chamber of Secrets. Voldemort was unaware that the Phoenix that Dumbledore named the Order of the Phoenix after pecked out the eyes of the basilisk that he named The Order of the Basilisk after.
The Order of the Basilisk was mainly controlled by Voldemort's council of Death Eaters. The Death Eaters were the only ones with any authority whatsoever. The only acception to this rule was named Wormtail.