What is a sidechain? How is it used for ducking, noise suppression, or bass emphasis?
Normally a compressor or other dynamic effect will react directly to the audio signal you feed it, the same signal that will then be compressed or processed. A side chain is an extra input that allows the dynamic processor to react to a different signal, allowing a wider range of possible effects on the audio signal at the output. Some sidechains have an external in/output jack, while others may only be connected to an internal parallel signal path.  
I've been struggling to come up with a decent analogy to help explain sidechains, and so far the best I've come up with is a "fake ID". Imagine a youth who wants to buy alcohol, but the local laws prohibit sales to people under a certain age. So the kid acquires a fake ID card that says he is of the legal age to buy booze. So for the purpose of the liquor store transaction, he is an adult; but when he drinks the liquor, he is still a youth. Now let's imagine the ID card says he is actually Itzhak Perlman or even Aretha Franklin, and this gets him or her into the VIP rooms of a fancy concert hall. He's treated like musical royalty, but he's really still the same youth, and he experiences all these different treatments as his original self, not as the person he's pretending to be.  
The most common way to use a sidechain is to take a Y split from the input signal and process it with an external device like an EQ. Then the EQ'd signal returns to the sidechain input of the dynamic processor, which then performs its effect on the original non-EQ'd signal. Again, the signal at the output has not been touched by the EQ, but the compressor/processor did its dynamic reaction to a copy of the signal that was run through the EQ.  
For example with a bass signal, the waves of the bass notes have much bigger signal peaks than the higher-pitched tones do, so those peaks trigger the dynamic response of a compressor much sooner than the rest of the signal would have, which may result in overcompression. This can be heard as muddiness, dull tone, or a loss of low end. If you use an EQ in the sidechain to turn down the lows a bit, then the peaks that hit the dynamic detector will be more even, the low-frequency peaks won't be higher than the midrange, so the compression will be more consistent, and you can set it accordingly. If you turn the lows in the sidechain down all the way, the end result can even work and sound like a bass expansion.  
As a switchable feature this system is included in a few comps like the Focusrite Compounder, dbx 166A/XL, ADK CLA1, and FEA Opti-FET. Their sidechain is hard-wired internally to an EQ which (in most cases other than the FEA) is preset for the lows at a fixed frequency. With these units there is no need to use an external processor or an in/output jack; these are examples of internal parallel signal paths.  
Many rack compressors feature a sidechain jack for use with an external processor. This processor doesn't have to be an EQ, it can be any sort of effect or gain control or whatever else. For example you could send your audio signal to a tremolo effect via the sidechain output of a compressor, and then that tremolo-pulsing signal would return to trigger the compression. The results may or may not be musically desirable, or worth the trouble, but it's good to know you have that kind of option available.  
Another technique is to feed the sidechain a completely independent control signal, like a miked drum, or a click track, or another instrument in the band. This allows you to have a great deal of creative control over how the processing will affect your audio signal. It's very popular in techno music to apply a heavy-squashing compressor to a bass track, and trigger it with a drum track. The same thing can be done with filters, gates, or any other sort of dynamic processing; and the control signal doesn't have to be rhythmic, it can be random. Some studio engineers will "fix" the timing of a musician who has poor rhythm by gating that musician's mixer channel, and triggering the gate from the channel of someone else in the band who has good rhythm. It's also fun to apply an envelope filter effect to vocals or sax for example, and trigger the filter with any other channel of the band mix. Get creative!  
Using an external trigger or processor means using a sidechain in/output jack on the dynamic device, typically a 1/4" TRS with the plug format "tip=send ring=return". In other words there is only one jack for both the send and the return, which means you need a special cable in order to do this. For sending your signal off to external processing and then returning, you need an "insert cable": a Y-shaped cable with a TRS (stereo) plug that splits off to two TS (mono) plugs. If you are using a completely outside trigger-signal source (like a rhythm track), then you'll take a balanced (XLR-TRS or TRS-TRS) cable and make sure that the signal output at the source device connects with the "ring" (return) connector of the sidechain jack. This may mean rewiring a plug at one end of the cable, swapping its tip and ring wires.  
One practical application used in broacast and PA systems is "ducking", where you have two audio channels and the volume of one is controlled by the volume of the other. A common example of this is the voice-over: music plays during a show or commercial, then an announcer or actor's voice comes on, and the volume of the music is lowered to make it easier to hear the talking voice. The music levels go up and down automatically whenever the controlling voice/audio passes the threshold. The voice-over channel goes into the sidechain return of the device that controls the level of the music; and after that processing the voice-over itself is then mixed together with the music channel.  
You can happily go your whole musical life without ever using a sidechain; but I will say that I use the one in my Opti-FET pedal all the time, to balance its response across the frequency range. It's a good tool to have!
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