The engines themselves were mostly Ion Drives.
Ion engines were fueled by power cells, liquid chemical reactants, onboard generators, or virtually any other device capable of providing sufficient power.
The engine utilized internal fusion reactions to produce a stream of highly-charged particles that were forced through the engine's exhaust port at nearly the speed of light. Ion engines were sometimes called ion rockets.
Ion engines were known to emit mildly radioactive byproducts, requiring onboard technicians to wear protective gear.
As far as fuel, mostly it was Hypermatter for capital ships/battle stations.
This included Death Star:
A grant from Republic Sienar Systems was given to the Republic Ministry of Science allowing them to continue research into hypermatter for planetary power, deep space mining, ... Some of this research was later used to refine the plans for Sienar's Expeditionary Battle Planetoid, which evolved into the Death Star project.
However, Hypermatter was not the only fuel: TIE fighters' ion engines "utilized internal fusion reactions to produce a stream of highly-charged particles that were forced through the engine's exhaust port at nearly the speed of light" (Source: Wookiepedia). The fuel for them was "High pressure radioactive gas" (see the tank for it in System Components list)