Considering that in The Force Awakens we see that Starkiller Base needs to drain a star to power it, what makes the Death Star so powerful? It's a giant superlaser, but what does it run on?
A Hypermatter Reactor. Just like Star Destroyers and other imperial vessels, but bigger (obviously).
Many of those present were researchers and scientists involved with various aspects of the battle station project — the drives, shield and tractor beam technology, hypermatter power plant — without a full understanding of what they were working on.
Catalyst - Chapter 8: Public Displays of Affection
The superlaser was fired by taking the power from the hypermatter reactor and focusing it all through the kyber crystals.