Normally a compressor reacts to the same signal it processes. With a sidechain, it listens to a different source instead. The classic example is a bassline compressed by the kick drum: each time the kick hits, the bass dips for an instant and then returns, so the two low-frequency sounds do not mask each other and each keeps its own space in the mix. The effect is often called ducking.
Set aggressively, sidechaining makes the gain pulse in and out in time with the kick, the rhythmic pumping heard in much house and electronic dance music. Jazz lo-fi uses it far more subtly, if at all: a slight dip of a pad or sampled loop under the kick can add gentle movement and clarity to the low end without drawing attention to itself.