In the 1977 film, the Rebel Alliance is able to destroy the Death Star by attacking its only weak point: a ray-shielded particle exhaust vent leading straight from the surface directly into its reactor core. Luke Skywalker (Mark Hamill) is able to successfully launch his X-wing fighter's torpedoes into the vent, impacting the core and triggering a catastrophic explosion. The explosion destroys the station before it can use its superlaser weapon to annihilate the Rebel base on Yavin 4.

Prior to its destruction and to mark the Death Star being fully operational, Governor Wilhuff Tarkin (Peter Cushing), who commanded the 'Planet Killer', had ordered the Death Star to destroy Princess Leia's (Carrie Fisher) home world of Alderaan in an attempt to press her into giving him the location of the secret Rebel headquarters. She gave them the location of Dantooine which housed a now-deserted Rebel base, but Tarkin had Alderaan destroyed anyway as a demonstration of the Empire's resolve to beat the rebels.

Luke Skywalker, Han Solo, Chewbacca, Obi-Wan Kenobi, C-3PO, and R2-D2 are pulled aboard the station by a tractor beam, where they discover and manage to rescue Princess Leia. As they make their escape, Obi-Wan dies duelling Darth Vader, enabling the others to flee the station. Based on the information from the escapees, the Rebel Alliancee is able to find a way to stop the Death Star. Its the exhaust vent!

More Info: en.wikipedia.org