Side chain compression is just a variation of well know compression effect.
The only difference between side chain compression and an ordinary compression is that at any moment in time in side chain the compression is driven by external signal (control) volume and applied to target (destination) sound rather then by the volume of the original target signal itself.
By order words side-chaining uses the signal level of another input to control the compression level of the original signal.
For sidechains that key off of external inputs, when the external signal is stronger, the compressor acts more strongly to reduce output gain.
This small difference plays huge part. the side chain effect is used everywhere we one need to lower music/instrument level when other instrument is played. For example it is used to lower bass volume when kick drum hit to remove clashing frequencies and avoid uncontrolled boom or by disc jockeys to lower the music volume automatically when speaking; in this example, the DJ's microphone signal is converted to line level signal and routed to a stereo compressor's sidechain input. The music level is routed through the stereo compressor so that whenever the DJ speaks, the compressor reduces the volume of the music, a process called ducking.
Side chaining opens all sorts of possibilities. One can heavily EQ the control signal so only when selected frequencies appear in the control signal, compress kicks in. Such a compressor can be used as a de-esser, reducing the level of annoying vocal sibilance in the range of 6-9 kHz. It can be assembled from a standard compressor and an equalizer by feeding a 6-9 kHz-boosted copy of the original signal into the side-chain input of the compressor.