The lightsaber is powered by a Diatium power cell:
The diatium power cell was the primary power source for a lightsaber and one of the critical components of a lightsaber hilt. Located directly below and wired into the primary crystal mount, to which was affixed the weapon's primary focusing crystal, these types of power cells were in turn wrapped within the power field conductor, power vortex ring and an inert power insulator. The lightsaber's pommel cap and its recharging port were attached directly to the diatum cell.
Lightsabers were initially (as protosabres) powered by a belt-mounted power pack connected to the saber via a power cable:
It seems the Dark Lords of the Sith Empire were ultimately responsible for the advancement of lightsabers, replacing the belt-mounted power pack with a power cell within the hilt. An internal superconductor was introduced, which transferred the returning looped energy from the negative-charged flux aperture back into an internal power cell. With this modification, the power cell would only expend power when the energy loop was broken, such as when the lightsaber cut something, solving the power supply problem.