A side-chain is an auxillary input on a compressor that allows another signal to control the compressor. In other words the main signal's dynamics are controlled by the dynamics of the signal that is input to the Side Chain. There's a whole lot of uses for it but a common one would be as a de-esser for vocals. Patch the vocal through an equalizer and boost the hi-freqs (8-12K). Take the output of the equalizer into the Side Chain input of the compressor. As the singer sings the compressor will act normally until he hits a syllibant sound. At that point the compressor will clamp down on it harder because the signal at the Side Chain has it's hi-frequencies boosted.
You ever been in a restuarant with the music playing at a real nice level and the hostess announces the next party to be seated and the music drops to the background while she has the mic keyed? That's ducking. Wally World does the same thing when they come over the speakers and tell security to videotape the red area. Although the restaurant and Wal-Mart will most likely use an automated type mixer (not like a recording mixer it's just a rack unit deal), a compressor with a Side-Chain input can accomplish the same thing.
Generally, you can set how quickly the main signal (which is the one that is ducking) responds to the overriding signal (the mic or whatever) and you can also deep it ducks. Wal-Mart kills the background music while your restaurant may duck it just enough so the voice is predominant. Another way to use it might be that you want the guitar solo to stand out. Input the guitar solo to the Side Chain so that when it cranks up, the rest of the mix "ducks" slightly.