
Most people, when they think of side-chaining, consider two main uses; there's the dubstep and jungle classic, of ducking the sub-bass as mentioned above; and then there's the "French House" staple, beloved of the likes of Daft Punk, in which you help create the world's biggest kick drum by sidechaining almost everything off the kick. Synths, strings, sample loops, the works. It creates a strong "pumping" sound and gives your mix movement, although it's not exactly innovative these days. But it certainly works.
To do this, you simply need to set your compressor's side chain input to take a signal from your kick drum. However, this method can cause problems - for example, if you want the "pumping" sound to continue when the kick is not playing - so in this case, it's useful to set up a new channel with no audio output. Then put a kick drum on this channel, so you can have full control over when the compressor is triggered or not.
That's the obvious use, but what else can you do with a compressor and a side-chain input? Well, if we consider that the main effect is to make something get out of the way of something else, it becomes extremely handy in mixdowns. A very common area for treatment is the vocal. This will normally be one of the most important features of your track, so it needs to come through clearly. As such, it's worth setting up a compressor on pads, strings and synths, and sidechaining that off the vocal. Nothing extreme - just take off 2 or 3 dB, but it will help make space for your vocal.
Or take effects tails - reverbs, or delays. These are great for giving space to your sounds, but aren't really featured sounds in themselves. By gently sidechaining them off your drum buss, or synth line, you can make sure that they simply add atmosphere, and not mud.

These are a few tricks - but they're just the ones that involve compressors, and there are a lot of other effects that can make use of a side-chain signal. The other main dynamic effect would be a noise-gate. These can be a great way of making a dull synth chord sound much more vital - try gating your synth and then setting the side-chain input to something like a hi-hat pattern, or a drum buss. The effect will be a stuttered, rhythmic synth pattern. Or you could lock a channel down to your main groove - for instance, sync a pad sound to your kick to glue the rhythms down and tighten up the pattern.
Similarly, you can create crazy new rhythms with a side-chain filter. Put the filter on a vocal, bass or synth sound, and set the side-chain input to something rhythmic, though not too detailed - a snare or simple bongo pattern. Then you can tweak the attack and release of the filter to make whole new sounds, movement or just more interesting tones on your original track.

As you can see, side-chains can be used across the board - from useful mixdown tools to help give subtle tweaks to your instruments, all the way to sources of creative visions that can help you come up with riffs, sounds and ideas you'd never have thought of otherwise. So load up some of those boring old plugins, and start combining your sounds for maximum inspiration!