The Alliance is a Colonial term used to describe those under the subjugation of the Cylon Empire (TOS: "Saga of a Star World").
The Cylons, as to be expected, govern the Alliance. Members of the Alliance often relinquish freedoms and rights to the Cylons in return for their continued existence. They also work for the Cylons in some capacity.
For instance, the Ovions are members of the Alliance and operate a facade chancery on Carillon. The Ovions also mine tylium from the planetoid for the Cylons to use in their crafts. An Ovion reports to a Centurion when all the warriors appear to be gathered at Sire Uri's awards ceremony, prompting the Cylon Attack (TOS: "Saga of a Star World").
The Thetas are also members of the Alliance. Created by Dr. Ravashol, they help Dr. Ravishol develop his inventions and perform necessary labor on Arcta. They rebel against the Cylons, however, and help the Colonials destroy the Ravashol pulsar when Dr. Ravashol asks them to do so (TOS: "The Gun on Ice Planet Zero, Part II").
In the separate continuity
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The Alliance is mentioned during Baltar's celebration toast to President Adar aboard Atlantia prior to the disastrous events at Cimtar, noting that no one before Adar had assembled the human race to speak as one voice before this organization (Annihilation!).