A thruster system relies on Newton's Third Law. This states that "every action must have an equal and opposite reaction."
Positively and negatively charged particles are separated. Using a set of magnetic fields, these particles are accelerated to the direction, opposite to the direction the starship or other body is trying to reach. Due to the impulse of these particles as they're expelled, a force acts on the body in the opposite direction, equal to the force with which the particles are expelled. As a result, the body is accelerated in this direction.
Charged particles are a side-effect of many quantum level reactions, such as matter-antimatter reactions. As a result, charged particles that can be used for the thruster systems can be found as a power core by-product.