The organization is at least hundreds of years old and most notably divided into two distinct groups, the Upper Moons and the Lower Moons. Each member is assigned a number from one to six (as well as an upper or lower status) and has this status carved onto their eye. It is unknown if the numbers are a ranking or able to be changed, since the former Lower Sixth (Kyogai) had the marks on his eyes covered with more scarring to show he lost his position when he was considered too weak, suggesting the previous scars could not be erased.
As Muzan is the progenitor of all demons and all demons naturally gain more power by ingesting his blood, the Demon Moons have the closest blood to him and apart from him are the most useful samples for studying demons. However, not any demon can take Muzan's blood and survive and an innate strength is required.
Though the Upper Moons have all held their positions for centuries and are responsible for the deaths of many Pillars in that time, the Lower Moons are considerably weaker and have often had to be replaced. With the death of Rui (Lower Fifth), Muzan summoned the remaining five Lower Moons and ultimately killed all but the First (Enmu) who was given more power and sent on a very specific mission to kill Tanjiro but he failed and was killed too, leaving no remaining Lower Moons. Though since Kyogai was left alive, it is likely other disgraced Lower Moons could still be alive.
The Upper Moons have yet to be introduced in full, though the Third (Akaza) showed up immediately after Enmu's death to fight the survivors and attempt to persuade the Pillar, Rengoku, to become a demon, suggesting he had tried it numerous times with other Pillars he ultimately killed. The Sixth (Daki) has been revealed to have a long history of using her power to rule a red light district (taking on a human form during the day and secretly killing those who oppose her) and has killed seven Pillars in the past.