The greatest challenge in designing the Death Star was not creating a cannon big enough to fire a beam that could destroy a planet, nor was it creating a battle station the size of a small moon. The greatest challenge was always powering a cannon big enough to fire a beam that could destroy a planet and moving a battle station the size of a small moon. The answer to both of these problems was solved with the invention of the hypermatter reactor.
The hypermatter reactor is the heart of the Death Star. The Death Star's hypermatter core is based largely on early Sienar Systems hypermatter implosion core that was the power source of the Confederacy of Independent Systems' Great Weapon (the early inspiration for the Death Star -- more on this later). Little is actually known about the details of the highly classified reactor design, but we do know that it is a massive fusion reactor fed by stellar fuel bottles that line the periphery of the main reactor chamber.
The Death Star's real-space propulsion system is made up of a network of ion engines that use converters to transform reactor power into thrust. The engine thrusters are primarily lined along the equator of the station.
Hyperspace travel is made possible with linked banks of hyperdrive field generators. Each bank contains 123 hyperdrive field generators. They are all tied together into one navigational matrix that is controlled from the overbridge.