The Death Star is a big starship that looks like a moon. It's main function is to keep order in the galaxy. All the Empire's top officals and military leaders live on the Death Star. Darth Vader also resides on the ship. It has a big blaster on the side that can destroy planets and moons alike.
The plans for the Death Star go back to Star Wars Attack of the Clones. During the battle of Genosis, Count Dooku is able to get the plans from a Separatist general to Chancellor Palpatine, or Darth Sidious. In episode III, Sidious releases the plans to his construction crew on the lava moon. During the time between episodes III and IV, the Death Star is completed and is ready to work. During the Battle of Yavin 4, the Death Star is destroyed.
The Death Star two was a bigger and stronger version of the first Death Star, but it was destroyed by Rebel forces.