Basically, something "from the side" affecting something else.

Sidechaining a compressor for example, to duck music out of the way for speech - You send music through a compressor, but send the vocal mic into the sidechain. When the announcer speaks, the compressor pushes the music out of the way.

If you're looking for a real-world example, listen to the DJ talent on almost any radio station.

Some other side chain uses.
EQing the side chain in a gating situation to impove the gate response.
Slide a duplicate region of an instrument back in time and using it to open a gate or trigger a compressor on the main track without any time delay. This works great for gating toms as you can make the gate open just before the inital strike.
You can also make a tempo based tremolo by feeding the side chain a click track so the gate on a guitar or whatever track opens and closes in time with the click.
Some engineers will even sidechain a bass compressor to the kick drum so that whenever there is a kick impact the bass is momentarily ducked which gives the kick more punch.
All in all, a lot of creative uses. Have Fun!!

Wow, very cool. I had no idea. Opens up a lot of possibilities. Thanks for your help guys.

very usefull for pumping dance tracks as well, put a bit of compression on to the bass and sidechain the kick, so the kick will "push through the bass" every hit. nice oomph oomph

oh, i am sorry just seen that dave62 covered that issue already, sorry
but have fun experimenting around a bit... it's very usefull

Suuuuuuper late to this party but just wanted to say thanks to the OP and the guys that answered! Great explanations and given me some ideas to have fun with!