The Galactic Empire is the creation and subsequent leader, a senator from Naboo known as Palpatine after he has taken control of the Senate at the end of Star Wars III - Revenge of the Sith. At his command are millions of clone storm trooper or even billions of Stormtrooper given the sheer numbers of planets that are under his control. Only when the Emperor has been killed will the Empire disolve. As there is to be a new trilogy, they'll be someone in the wings hoping to recreate the Empire to control the galaxy. The Empire is controlled from Coruscant which was also the seat of power for the Old Republic.
Although the Death Star was not originally built by the Empire to start with, the Empire assumed control of it. The Death Star was built by the Separatists to protect themselves but that failed. When the first Death Star was destroyed, they subsequently ordered a second much bigger and more powerful Death Star to be built in orbit round the forest Moon of Endor. It was only when the second Death Star which the Emperor himself was visiting did the Empire fall apart.
When Obi-Wan Kenobi killed General Grevious in a duel on Utapu, Palpatine put in motion the end of the Jedi Council and ordered his Stormtroopers to kill the remaining Jedi's calling them traitors. After the Jedis were killed off, Palpatine went to the Senate to declare the first Galactic Empire had been created.