Battle of Yavin
The Millennium Falcon arrived in the Alderaan system shortly after the planet's destruction and was promptly captured by the Death Star. Grand Moff Tarkin and Darth Vader determined that the freighter was attempting to deliver the Death Star plans to the Alliance. A tracking device was placed on the Falcon, which was then allowed to escape to accomplish its mission, in the hopes that the beacon would lead the Imperials to the hidden rebel base.
Princess Leia, who had been rescued from the Death Star immediately beforehand, suspected the freighter was being tracked, but without the time or resources necessary for a full evacuation of the base on Yavin IV, the Rebels decided to stay and fight.
An analysis of plans of the station (carried in R2-D2, an astromech droid) revealed one way by which the station could be destroyed. A small thermal exhaust port in one of the polar trenches led directly into the station's main reactor, and a proton torpedo fired down the shaft could detonate the reactor and destroy the station. Beam weapons would not work, however, as the shaft was protected by a ray shield. The Rebel pilots were less than optimistic about their chances, and doubted whether their targeting computers were accurate enough to hit the 2-meter-wide target, but a new recruit named Luke Skywalker assured them that the shot was perfectly achievable.
The Rebel's attack plan was for their X-wing fighters in Red Squadron to carry out an attack pattern near the station's equatorial regions to distract the Imperial forces, while the Rebel Y-wing fighters in Gold Squadron carried out the actual attack run. With the Death Star rapidly orbiting Yavin in order to position itself to destroy the Rebel base, time was of the essence.
Initially, the Imperial forces fared surprisingly poorly. Grand Moff Tarkin refused to deploy the station's TIE fighter squadrons, deeming it a waste of time and the Rebel attack a useless gesture. Their turbolasers -- designed to target capital ships -- were not fast or accurate enough to shoot down the Rebel fighters, and during this stage of the battle the Rebels only suffered one casualty due to a freak accident. Eventually Darth Vader, while not fully comprehending the reasoning behind the attack, decided to eliminate the Rebel fighters and ordered his personal TIE squadron to destroy them. This evened the odds, and the Rebels started sustaining casualties; by this stage however, Gold Squadron had located the trench and began their attack run.
Realizing what the trench was the primary Rebel objective, Vader decided to get involved in the fight himself and commandeered two TIE pilots to serve as wingmen. He soon intercepted Gold Squadron and used the narrow trench to his advantage, destroying the Y-wings before they even got a chance to fire their torpedoes at the exhaust port. Upon learning of Gold Squadron's failure, Red Leader decided to make the attack run himself, flanked by two other members of Red Squadron. Once again, Vader and his wingmen dropped in behind the X-wings and picked off Red Leader's escorts. This time, Red Leader had time to launch his torpedoes; unfortunately the Rebel's worst fears were proven true, as the targeting computer proved not to be accurate enough, and the torpedoes missed the exhaust port, exploding nearby on the Death Star's surface.
Before being shot down by Vader, Red Leader ordered Skywalker and the two remaining Red Squadron members, Biggs Darklighter and Wedge Antilles, to make one final run on the trench. Sure enough, Vader intercepted the remaining X-wings, damaging Antilles' fighter and forcing him out of the battle, then destroying Darklighter's craft. He was about to do the same to Skywalker when one of Vader's wingmen suddenly exploded. Han Solo, instead of fleeing the battle as he had previously intended, had decided to bring the Millennium Falcon into the fight. In the confusion, Vader's other wingman collided with Vader's fighter, killing himself and sending Vader hurtling away into space. Now with a clear shot, Luke decided not to use his targeting computer and instead relied on the Force to guide his shot at the exhaust port, sinking two torpedoes down the shaft. Skywalker, Solo, Antilles, and an unidentified Y-wing pilot (presumed to be Keyan Farlander) managed to fly away from the station before the torpedoes reached the main reactor, and the Death Star was consumed in a massive explosion. Darth Vader proved to be the sole Imperial survivor, and his hyperdrive-equipped fighter quickly left the system.
While the Alliance celebrated their victory, they also knew that the Empire would quickly return with as much force as they could muster, in order to avenge their defeat, and so began evacuating the base. Darth Vader returned with his new Super Star Destroyer, the Executor in a mission to capture the Rebel leaders, and would have successfully done so if not for another Star Destroyer squadron accidentally hyperdrive-ramming the Executor during a crucial point of the mission. Then, as the rebels moved their base to Hoth, they were ambushed in the Ison corridor. Thanks to the new A-Wing fighter, the rebels were victorious.
Alternate EU History
Bevel Lemelisk, the designer of the Death Star, was soon punished for the design flaw by being executed several times over until he designed a new Death Star that the Rebels could not destroy. This eventually resulted in the Death Star II.
Star Wars: Rogue One reveals that the exhaust port vulnerability of the Death Star was no accident; it was a "feature" carefully incorporated into the design by engineer Galen Erso, who had been coerced into working on the project.