The Rebellion also known as the Rebel Alliance was formed by members of the Old Republic who were in disagreement with Vice Chancellor Palpatine. Vice Chancellor Palpatine assumed the role of Galactic Emperor following the defeat of General Grevious. Leading members of the Rebellion are Mon Mothma and Princess Leia Organa. Only by killing the Emperor would the Empire be disolved. Throughout the original movies, the Rebellion are on the move, hiding on one planet (e.g. Yavin IV) then moving to another (e.g. Hoth) when they are discovered.
In star Wars - The Force Awakens, the Rebel Alliance have changed their name to simply the Resistance. The aims of the First Order is still the same, control of the galaxy but with a different name so because the Empire their name then so does the Rebel Alliance. The leader of the Resistance is General Leia now and based on D`Qar.