The Alliance formed after Supreme Chancellor Palpatine transformed the Galactic Republic into the Empire with himself as Emperor. According to the role-playing game, resistance was relatively quiet until a military ship commanded by Tarkin was blocked by protesters who stood on the ship's landing pad and refused to move. With implied permission from the Emperor, Tarkin landed the ship anyway, right on the protestors, injuring and killing many.
Senators Mon Mothma and Bail Organa finally realized the Empire had absolutely no regard to the rights, or even the lives of its citizens and normal political channels were useless to address the problem. With their help, an underground organization was formed to plan the removal of the Emperor, but the growing ruthlessness of the state forced them into initial secrecy.
When they were ready to fight, it became clear that the cause needed legitimacy with something public. Mon Mothma supplied an open Declaration of Rebellion in which the grievances of the Alliance were stated and concluded with an open threat to depose the Emperor. With that act, the Empire went to work to crush the insurrection, while various allies secretly made their contributions to the Alliance.
The most notable equipment contributions include the defection of the Incom Corporation staff and all relevant material involved in the development of an advanced spacefighter, the X-Wing. Later the fish-like species called the Callamaria joined the rebellion and used their renowned skill in ship construction to supply badly needed capital ships that could take on the Imperial Navy.
According to the RPG, part of the Alliance's strategy was the doctrine of space denial. That is, the alliance would attack Imperial shipping in hit and run raids both to disrupt Imperial supplies and operations and to confiscate desperately needed materials. In that regard, their fighters had hyperdrive capability along prearranged routes which would allow this kind of harassment and escape before the Empire could react.
Through their intelligence network, the Alliance learned of the construction of the Death Star, an enormous terror weapon intend to secure the Empire's power. An operation headed by Princess Leia was successful in retrieving the schematics of the station, but Darth Vader discovered the act and captured Leia. However, in the events described in the film, Star Wars: A New Hope, Luke Skywalker and friends managed to free the Princess and deliver the plans to the Alliance. This not only allowed the Alliance to destroy the station in The Battle of Yavin, but also in Luke, they gained their only loyal member with training in the use of The Force and the skills of the purged Jedi Knights.
The Alliance suffered a setback when their main base was overrun by Darth Vader's forces on Hoth, but eventually they later managed to mass a fleet to destroy the second Death Star in the Battle of Endor in which the Emperor -- or one of his clones -- was killed as well.
With the fall of the Empire, the New Republic was established with Mon Mothma as Supreme Chancellor in conjunction with the restored Senate.