In both cases, the engines work by shooting negatively-charged ions out the back at extremely high velocities. In Star Wars, this idea works really well; in reality, this is really slow because you're moving by throwing out single atoms.
Ion engines are extremely efficient propulsors in a vacuum (they have a high Isp), and therefore are useful for applications such as deep space probes, satellite stationkeeping, and interplanetary travel in general! One reason they are efficient is that the energy used is supplied to the engine in the form of electricity. This means you can use solar power or even nuclear batteries to provide power, which is much more efficient in terms of mass and space than carrying fuel. The only consumable you need to carry for thrust is the ion particle supply, usually xenon or some other inert gas. You can conserve fuel by using a more powerful electric field in the engine; a particle tossed out the back twice as fast yields twice as much thrust for the same rate of flow of particles as per the equation:

T = v(dm / dt)

Thrust is equal to the velocity of the ejecta times the rate of mass flow (delta-M over delta-T). So doubling the velocity gives you twice the thrust for the same fuel consumption. The ion engine is efficient because it allows you to increase your thrust by the input of a potentially renewable energy type, in this case electricity.

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