Normally, a gate or a compressor / expander will change the amplitude of a signal depending on what the amplitude of that same signal is. That is, the signal that comes out depends only on the signal that comes in. A compressor, for example, could take the lower amplitude portions of the signal and make them louder or take the louder portions of the signal and make them quieter. A side chained compressor uses two signals. One signal is modified by the compressor and one signal controls the compressor.
A side chained compressor is a compressor that modifies one input signal based on the amplitude of another input signal.
A side chained compressor could, for example, make the modified signal quieter if the controlling signal is loud.
A side chained compressor is interesting as it allows the dynamics of various instruments in the same song to come together. For example, you can put a side chained expander on the bass using the drums as the control signal. The bass will then thump well with the drums.