The Patriarchs are a group of Christian individuals who rule the terran world of Bountiful. As they are only mentioned in the memories of neurally resocialized marine Ardo Melnikov, it cannot be said for certain that the group actually exists, or existed as the case may be as in the same memories, Bountiful was invaded by the zerg. In all likelihood however, the Patriarchs do exist.
The Patriarchs have their origins in the early 23rd century, a time where the United Powers League was imposing its rule upon Earth. Part of its agenda was the surpression of many of Earth's religions, Christianity among them. Refusing to comply, the Patriarchs were banished to the Koprulu Sector along with those that had been rounded up in Project: Purification. Most likely they were loaded onto the Nagglfar.
With the re-development of warp engines by the terrans of Tarsonis in the late 24th century and the re-establishment of contact with similar minded individuals on Moria and Umoja, the Patriarchs sought to establish their own colony, built and shaped to fit their own ideals. Such a colony was the world of Bountiful, many like-minded individuals following in their wake. Since then, the Patriarchs have ruled Bountiful.
The Patriarchs, as the name implies, are a male-dominated organization. Most, if not all of its members are gray haired elders. They are involved in both the administration of Bountiful and the education of youngsters, both in Christianity and regular subjects. Although they have a clear stance on certain issues (eg. were against makeup and other cosmetics), it does not appear that they went out of their way to enforce their beliefs (as people settled Bountiful willingly, they would not have to enforce their abilities at all).
Women can be said to be part of the organization, in the form of "Sisters". They appear to be subordinate however, as there is no known Matriarch.