Short answers, pulled from the story.
Senator Leia Organa founded the Resistance, breaking away from the New Republic's military with a group of fellow Rebel Alliance veterans. She formed it as a precaution against the rising First Order, a threat the New Republic refused to take seriously.
The First Order is the military dictatorship the Resistance was created to oppose. It grew out of the fallen Galactic Empire's remnants in the Unknown Regions and repeatedly violated the Galactic Concordance by rebuilding old Imperial fleets and stormtrooper forces.
The Resistance was introduced onscreen in the 2015 film Star Wars: The Force Awakens. It went on to appear in Star Wars: The Last Jedi in 2017 and Star Wars: The Rise of Skywalker in 2019.
The Resistance's Fleet Command is led by Fleet Admiral Gial Ackbar, General Leia Organa, and Vice Admiral Amilyn Holdo. Holdo assumes command after Ackbar is killed and Organa is seriously wounded in The Last Jedi.
General Caluan Ematt and Commander Larma D'Acy lead the Resistance's ground forces. Because the Resistance is too small to support dedicated infantry units, its pilots and technicians are trained to fight on the ground when necessary.
Under the joint command of Poe Dameron and Finn, the Resistance fleet travels to Exegol to confront the Sith Eternal's Final Order and wins. After Palpatine's death, citizens across the galaxy rise up against the First Order, bringing the regime to its end.