Chain "Stretch" [courtesy of sheldonbrown.com]

Cyclists often speak of chain "stretch", as if the side plates of an old chain were pulled out of shape by the repeated stresses of pedaling. This is not actually how chains elongate. The major cause of chain "stretch" is wearing away of the metal where the rivet rotates inside of the bushing (or the "bushing" part of the inside plate) as the chain links flex and straighten as the chain goes onto and off of the sprockets. If you take apart an old, worn out chain, you can easily see the little notches worn into the sides of the rivets by the inside edges of the bushings.

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To prevent chain 'stretch' one needs to take certain simple considerations:
1. Do not over tension your chain. I understand that it's important to prevent chain drop but having your chain too tight will drastically increase chain wear.
2. Clean and relubricate chains often, consider appropriate lubrications for given weather conditions (wet lub for wet weather, wax for snow, dry teflon for dry weather). Dirty chains hold onto dirt and road grime which helps to eat away at the chain's rollers.
3. Straighten your chainline. A crooked chainline wears a chain in much the same way that cross-chaining on your geared bike.

I've still yet to see a customer actually break a chain. If it happens, I imagine they happen under very extreme conditions (world class sprints doing jumps on 90inch+ gearings). Cleaning your chain and running a little slack not only improves your chain's life and performance, it also helps prevent cog and chainring wear.