The Battle of Krypton was a civil war that occurred after General Dru-Zod orchestrated a coup in an attempt to take control of Krypton from the Council of Five. It was Zod's attempt to correct Krypton's leadership in hopes of saving his planet from further destabilization and, ultimately, destruction, but the actions of the Sapphire Guard neutralized the Sword of Rao after trapping Zod, ending the war.
During the Kryptonian Expansion, artificial population control was implemented on Krypton, leading to the space exploration program being abandoned as genetic cultivation was further explored, exhausting Krypton's natural resources. In response, the Council of Five authorized the mining of Krypton's core. Because of this, the mining caused Krypton to implode.
General Dru-Zod, leader of the Warrior Guild, formed the Sword of Rao and was joined by his subordinates. His influence within the Warrior Guild also led many soldiers to join in, although many didn't, being loyal to the democratically elected council.
Jor-El went a different way: knowing that the population of Krypton was genetically bred, each for a specific role in society, he knew that any change to the order would spell massive chaos, thus coming to the conclusion that everyone on the planet was dead already, way before the planet's implosion. He and his wife, Lara Lor-Van, decided to gain access to the growth codex, the registry that contained the genetic information of every individual Kryptonian, and to try to bear a child for the first time since the implementation of the population control. Jor-El's plan was to use the codex and infuse it into his child, then send the child away to a planet with a yellow star in Sector 2814 (Earth). The first part was successful. Lara and Jor-El managed to conceive a child: Kal-El. However, at that time he wasn't able to legally gain access to the codex, thus he went to the council in an attempt to do so.
Coup and war
Jor-El's warning and plan
Jor-El went to the council with his warning and plan. The council, skeptical of this, decided to deliberate again, much to Jor-El's dismay. At the same time, Zod and his followers launched the coup to save Krypton. Zod announced that the council is disbanded, killing High Eminence Ro-Zar and arresting the other council members. Seeing Jor-El, Zod was excited as his old friend had agreed that the planet was dying, however, Jor-El did not agree with Zod's plan for keeping the bloodlines, thus disappointing Zod. Still considering Jor-El a friend, he decided to take Jor-El away while he moved to other matters, namely to consolidate his influence as a new leader.
Outside, his forces clashed with loyalist forces, seemingly on equal standing. On Kandor, chaos ran free as capital ships fought in the skies and assault ships ruined Kandor. Jor-El escaped captivity, saw the chaos, and decided to act on his own. Grabbing his war kite H'Raka, he went to a genesis chamber where the codex was located and stole it to infuse the genetic information into his son. Zod knew of this and sent two attack ships to intercept him, though Jor-El managed to escape back to his citadel.
On Kandor, Zod's forces were being defeated as reinforcements came to join the loyalists from other parts of Krypton. During this, Zod led a squadron to Jor-El's citadel to arrest him. 
Meanwhile, Jor-El managed to infuse the codex into his son and prepared a ship to send him away, while also including a command key for use in the Kryptonian scout ship on Earth. Informed of Zod's approach by Kelex, his service droid, Jor-El readied his battle armor to face him. In the citadel, Zod knew what had been done, and in shock found out about the natural birth, which he considered heresy. Finally losing patience with his old friend, Zod engaged Jor-El in combat. Zod, being the leader of the Warrior Guild, was the better fighter, however Jor-El grew up alongside him and also learned to fight, and in this fight, Jor-El managed to hold his own and defeat Zod in hand-to-hand combat. Defeated, Zod pleaded with Lara to cancel the ship's departure, but failed to persuade her, and the ship began its escape from Krypton. Zod, enraged, used his blade gauntlet to kill Jor-El, then went out and ordered a ship to destroy the escape pod. In the last minute, a loyalist capital ship destroyed Zod's ship and major reinforcements from the loyalists managed to end Zod's coup attempt and defeat the Sword of Rao. Zod's insurrection had failed.
After the coup attempt, the law council regained control over Kandor and Krypton while also sentencing the Sword of Rao for their crimes. As it was the most heinous and high-profile crime committed on Krypton for centuries, the council saw it right to put the war criminals through 300 years of somatic reconditioning in the Phantom Zone. Before being sent to the Phantom Zone, Zod declared to Lara that he would find Kal-El, no matter where he was sent. Afterwards, all the surviving war criminals were put in cryogenic capsules and sent onto the Black Zero for their trip to the Phantom Zone.
Not long after, Jor-El's prediction came true; Krypton imploded. Lara could only agree in vain with Jor-El and hoped that Kal-El would avoid the same mistakes Krypton made as Krypton's core gave way and the planet buckled, killing all life. As the planet imploded, the ripple effect of its destruction broke the boundaries of the Phantom Zone and freed Dru-Zod and the Sword of Rao in the Black Zero. Distraught at the remains of their dead world, Zod and his allies reconfigured the Black Zero into a starship and began searching ancient Kryptonian outposts for life, but found nothing but death. During their 33-year search, they scavenged armor, weapons, and a world engine with which they might rebuild Krypton, until one day, they received a distress beacon from an ancient Kryptonian scout ship activated by Clark Kent which led them to Earth.