Volvo's claiming their new flywheel design will be able to give a four cylinder the equivalent of six cylinder power as well as save 20% of combustion energy.
A flywheel is just a name for a rotational device that's able to store energy. It does this by using inertia and saving rotational energy. The amount of energy that can be stored is based on the moment of inertia and rotational speed, where moment of inertia is based on mass and radius of the wheel. Typical flywheel designs might include differing weights at certain points in the wheel that help keep the rotational speed more consistent and higher.
In your typical automotive application, a flywheel sits on the end of your crankshaft and converts the energy and movement of the crankshaft into a more consistent rotational speed.
Volvo is claiming that their flywheel will be able to store so much energy that when the engine is idling it can actually turn off and the energy stored in the wheel is enough to start combustion again. If so that could explain their 20% fuel savings number as engine idling is a well known waster of energy (why electrical cars and the Prius shut off at a stop). The flywheel also sounds like it is using aspects popular in hybrids and electric cars: regenerative braking. There've been a lot of attempts to incorporate regenerative braking into designs, the idea that you can get back part of the energy you're wasting to brake. But generally these designs haven't been efficient enough to be practical (plus once you add complications, you add more room for failure, and the more components the heavier your car thus reducing your energy savings).
They're supposed to start field testing this sometime later this year.