It's simply a watch that's kept in sync with the US atomic clock in Boulder. The precision is based on the oscillation frequency between a Cs133 nucleus and its electrons. That's the main difference between it and your standard fare wall clock really. There's no more accuracy to be had than with an atomic clock.
The solar variety is just powered by sunlight.
And before anyone asks, no, it's not radioactive (133 is a stable isotope of Cesium). Oscillatory motion and nuclear decay are entirely separate phenomena. As a matter of fact, radioactive decay would gradually increase the error over time, depending on the mechanism.
Hope that helps.

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