In as small amount of words possible, Sidechain is the process of using the output of one track to control the action of a compressor on a completely different track.
In more Lehmann's terms, if you have something to trigger the sidechain (a kick, snare, synth, ANYTHING), whenever that trigger is hit/played, a compressor will affect another track.
For example, its most common usage is the kick/sub bass combo. A kick's main frequencies are between 80-120Hz (depending on the kick), and a sub bass is usually between 30-80Hz. Therefore, if both play at the same time you will have a clash. Usually it results in distortion, clipping, and all round shittyness. So, if you add a compressor to the sub bass, switch on the sidechain function, then make the kick drum the trigger, you will have the sub bass 'ducking' every time the kick hits. The sub bass will compress down (depending on the compressors settings) to make room for the kick so that there is no clash of frequencies.
There are plenty of other uses for it as well. E.g. reverb ducking on synths, pumping feeling, vibrato effect. I wont bore you with those though
