The foundation for the Rebellion started as far as near the end of the Clone Wars when a series of like-minded politicians and nobles including Mon Mothma, the Organa family, and Garm Del Ibis started talking about starting a petition to restore normal democracy function to the Republic to present to Supreme Chancellor Palpatine. Of course, this did not happen as the Empire was formed and the Senators, while sharing the same vision and keeping in touch, kept their mouths shut out of fear of retaliation by Imperial agents.
The Rebellion was truly founded when a Dark Jedi named Galen Marek (who had been secretly trained as a Sith Apprentice codenamed "Starkiller" by Darth Vader) decided he would help Vader destroy the Emperor's enemies by gathering them all together. However, Vader's brutal discipline, constant lies and betrayals, and a fit of consicence brought on by Marek's continued interaction with his pilot June Eclipse and the Jedi Rohm Kota caused him to decide to genuinely help the Rebellion be formed
The Rebellion leaders were soon captured by Vader and his agents after the Rebellion was formed in a frost-heavy region of Corellia, Vader took them to the Death Star. Marek began a daring rescue mission of them on the Death Star, badly damaging the somewhat-finished superstation along the way. He succeeded in saving them by using his incredible Force powers to hold off both Palpatine and Vader at once but had to sacrifice his own life to do so in order for the Rebellion leaders to get away safely.
Later on Kasshyyk, the Rebellion leaders finalized the Alliance by using the Marek family insignia as their flag.