In the field of Physics, Destructive Interference usually refers to the Interference of waves. Destructive Interference basically refers to the tendency of two opposite waves cancelling each other out. Sound waves are a good example, and you can see Destructive Interference in action if you have a tuning fork around.

Basically, when you whack the fork, the noise coming off the flat faces of the two prongs is much louder then the noise coming from the corners of them.

I'm not entirely sure how this works, because I came really close to failing physics. But I've seen (or rather heard) it.

This is actually being used (well, prototyped right now) in really new cars - basically, you have large speakers under the seats, and then microphones pick up noise coming into the car. The speakers then generate the opposite sound wave, and the two cancel each other out for a totally silent trip.

The only other thing I recall is that my teacher said it should be possible for airplanes to mask themselves from radar with destructive interference like this. Radar as mostly based on echoing sound waves, so if the plane had a radar detector that identified the frequency the radar was using, it could generate the opposite wave and cancel the radar beam out as it comes towards it. I guess the receiver would still get a blip for a second, but it would at least distort it after that.
Destructive interference is a truly amazing concept involving the properties of waves. I'll try to explain it with a little more detail than the previous entry.

Put simply, destructive intereference occurs when two waves of the same wavelength cancel eachother out. This could can include sound, water waves, and even light which has been proved to have the properties of a wave through Young's Experiment.

To accurately describe the phenonemon, first imagine a long piece of taut string in front of you. Now imagine the crest of a wave heading down the the string. Now if this crest moving down the string was all there was to it, it would just go on until infinity (in ideal conditions) or until the string ended. Now imagine a trough of a wave of the same wavelength whipping along in the other direction heading towards the crest. When the crest and the trough meet, for a split moment the string would appear absolutely taught with no waves on it. This is called destructive interference. And when waves hit eachother and create desctructive interference, they are said to be out of phase. If these waves were soundwaves, a listener would hear nothing if they were positioned at that exact point

The ramifications of this concept are mind-boggling. One is actually able to cancel out sound! The idea is used in many of the products you see today. Bose makes a headset that actually cancels out incoming sound. The way it works is quite simple. It has a microphone on the outside that takes sound in, then a processor inside the unit produces the same wave, but out of phase, out of the headset. The effect is absolute silence for the wearer of the headset. You can see these things in action every day at university libraries across the country.

Auto makers are using the idea in their cars now for the absolute quiet ride you keep hearing about. Microphones on the outside take in sound and processors broadcast the sound out of phase on the inside of the car. The previous post is partially correct in that planes can cancel out radar, but radar does not technically consist of soundwaves. The waves used are out of the human hearing range. If they weren't, trips to the airport would be quite a bit more noisy. And don't get me started about driving by the local TV station!