Birth and Early Life
Leia was born in the medical facilities of the Tantive IV shortly after Obi-Wan Kenobi defeated Darth Vader in a lightsabre duel on Mustafar. Fearing to have the children raised by their father, Obi-Wan, Yoda, and Bail Organa decided to separate the twins and conceal their identities. Bail Organa took Leia to his home on Alderaan to raise as his own adopted daughter.
Rebel Against the Empire
Under the rule of the Galactic Empire, Leia acted as a diplomat on behalf of Senator Organa. Secretly she was a courier for the Rebel Alliance, carrying messages among the various resistance cells. In this capacity, she came into possession of the technical schematics for the Death Star, which she intended to deliver to Senator Organa on Alderaan. Her ship was intercepted near Tatooine, where she transfered the plans to R2-D2 and sent him to the planet in an escape pod before being captured by Imperial Stormtroopers.
Darth Vader moved Leia to the Death Star for interrogation, but she proved exceptionally resistant to Imperial interrogation methods. Grand Moff Tarkin attempted to coerce her to reveal the location of the Rebellion's secret headquarters by threatening to destroy Alderaan, and she attempted to save the planet by providing the location of a base that the Rebels had already abandoned, only to have Tarkin destroy Alderaan anyway, before even checking the truth of her answer.
Leia was rescued by Luke Skywalker, Han Solo, Obi-Wan Kenobi, Chewbacca, R2-D2, and C-3PO, who were themselves attempting to escape the Death Star after the Millennium Falcon was captured when trying to reach Alderaan. She lead the group to the true location of the Rebel base, the fourth moon of the gas giant Yavin. Knowing the Imperials were probably tracking the Millennium Falcon, the Rebels quickly analyzed the Death Star's schematics and discovered a way to destroy it. Following the battle, they quickly evacuated their base.
Leia remained a high-ranking member of the Rebel Alliance. At the Battle of Hoth, she was separated from most of the Alliance leadership and forced to flee with Han Solo in the Millennium Falcon. During the flight from Imperial forces, she developed a romantic relationship with Solo. Both were eventually captured when they stopped at Bespin to repair the Falcon's hyperdrive. Luke attempted to rescue them, creating enough confusion for Leia to escape, while Han Solo -- cryogenically frozen and encased in carbonite -- was taken to Tatooine by Boba Fett.
Leia participated in the raid on the palace of Jabba the Hutt on Tatooine which freed Han. She, Han, and Luke then led the commando team that would attempt to destroy the deflector shield protecting the Empire's new Death Star, which was still under construction in orbit over Endor.
General of the Resistance
Sometime after the Battle of Endor and the defeat of the Empire, remnants of the Imperial military regrouped to form the First Order, which continued to hold territory in spite of the formation of a New Republic. Under the leadership of Supreme Leader Snoke, the First Order began attacking the Republic in an effort to regain control of the galaxy. For unspecified reasons, the New Republic was unable to counter this threat directly, so Leia formed a proxy organization called the Resistance to keep the First Order at bay. As of Star Wars: The Force Awakens, Leia is the commanding officer of the Resistance, holding the rank of General.
At some time after the Battle of Endor, her relationship with Han Solo resulted in the birth of Ben Solo.
After the Battle of Endor, Leia remained a prominent figure in the New Republic. She received Jedi training from her brother, and married Han Solo and bore him three children: twins Jacen and Jaina Solo, and Anakin Solo.