The Eastern Alliance is a fascist party committed to the complete rule of Terra and her colonies. It is comprised of humans who may not have been descendants from Colonials. The connotation is that they are somehow related, most likely indirectly, to the Thirteenth Tribe.
The Eastern Alliance not only believes that it is the strongest force in the galaxy, but also that it has a manifest destiny to rule all life in the universe, a task they were "created" for. It also believes that life falls under two classifications, the strong and the weak, and that this is an immutable fact (Baltar's Escape).
In pursuit of this so-called destiny, the Alliance is responsible for committing war crimes on a large magnitude and was embroiled in a bitter war with the democratic Nationalists, whom they view as weak. Furthermore, they planned a massive assault against the Nationalists while negotiating with President Arends on a peace treaty in much the same way that the Cylons and Baltar convinced the Colonials that they were on the cusp of peace. In preparation for their attack, the Eastern Alliance moved its high ranking military personnel down in underground bunkers, leaving the world above unaware of their plan, so as to solve their overpopulation problem.
Like their enemy, their technology is below Colonial standard and are not likely able to utilize extensive star travel as the Twelve Colonies had prior to their destruction ("Baltar's Escape", "Experiment in Terra").
After encountering the sleeper ship with Sarah Fowler and Michael, the Fleet (notably the crew of the Galactica) hears of the brutality of the Alliance. After allowing the sleeper ship to leave, Starbuck and Apollo (who provide escort to the unarmed ship) encounter enforcers from the Alliance on the planet Paradeen -- a planet whose inhabitants were destroyed by a biological attack unleashed by the Alliance. This encounter allows the Warriors to understand the limited technology the Eastern Alliance utilize; Commandant Leiter, commander of said craft, is astonished by the size and magnitude of the last surviving Colonial battlestar (Greetings from Earth).
Later on, Sire Domra and the Quorum of Twelve attempt to broker a peace with the Eastern Alliance, believing that the Alliance patriots have been mistreated by the Warriors. This fig of peace is later broken, as eleven of the Quorum members walk into a trap, instead of a peace conference. Only then is the council willing to reinstate Commander Adama's full authority over the Fleet, as well as his own ship. While Leiter's destroyer escapes (Baltar's Escape), the ship is tracked to one of Terra's lunar colonies.
Galactica and her fleet discover Terra. Instead of establishing contact, a Viper patrol is sent on reconnaissance. However, a Being of Light intervenes during this mission and gives Apollo the ability to appear as the lost Nationalist Colonel Charlie Watts. Due to Apollo's odd lines of questioning and behaviors, he is almost unsuccessful in thwarting the Eastern Alliance's plan to commit genocide upon the Nationalists in an all-out thermonuclear attack. After Galactica destroys the inbound warheads, the Alliance reconsiders its fight with the Terran Nationalists (Experiment in Terra).