Way back in December I posted this picture when I was ranting about how much I hated glitter.

I mentioned that I would eventually run a post on the picture and I finally got around to it. Here is the same picture, but with a nifty red arrow meant to direct your attention.

My commute is flat and I pretty much ride in the same gear all the time. The only hills that really confront me are stiff headwinds. Around early December last year my chain started to jump around like kid on Christmas morning each time I leaned into my pedals to climb a slight incline or to pedal into the wind. At first I thought that all that was needed was a simple derailleur adjustment. After that did not work I did what any person with a bike related issue should do - I visited Sheldon Brown's site. Thankfully some kind souls have kept the site up after his unfortunate passing.
I have completely ignored chain life over the course of my cycling life. I do not know if I am lucky, but I have had the same chain on my road bike for 17 years and have only oiled it to keep it up. I have had no shifting problems on that bike, even after putting over 20,000 miles on it. My luck did not extend to this bike though. As can been seen in the picture, one sprocket is distinctly different than the rest. This is a result of "chain stretch."
Sheldon's site does an exemplary job of explaining the fact that chains do not really stretch, rather the movable parts of the chain wear out and thus change the way the links of the chain interact with the cassette and chain rings. One result is what you see in the picture, the teeth of the sprockets and chain rings can change shape. When the teeth wear off, the chain can slip when pressure is applied.
Generally speaking it is easier and cheaper to replace a chain as opposed to a cassette and thus it is a good idea to keep an eye on the condition of your chain. It is good to lube and clean your chain, Sheldon's site is a great resource for this, and to measure your chain to get a sense of the "stretch" that is going on over time. This can be done with a specific tool or with a tape measure. Here is a portion of text from Sheldon's site:
The standard way to measure chain wear is with a ruler or steel tape
measure. This can be done without removing the chain from the bicycle.
The normal technique is to measure a one-foot length, placing an inch
mark of the ruler at the side of one rivet, then looking at the
corresponding rivet 12 complete links away. On a new, unworn chain, this
rivet will also line up exactly with an inch mark. With a worn chain,
the rivet will be past the inch mark. [For accurate measurement, the
chain should be held under some tension -- either on the bicycle, or
hanging. Also, use a metal ruler or tape measure. Wood, plastic and
cloth all can expand or shrink.-- John Allen]
If you do visit Sheldon's site, be prepared to have time fly quickly. It is pretty easy to get engulfed in all of the great content and soon you will wonder where the time has gone. Moral of the story here is to keep an eye on your chain wear, or you might find yourself having to replace more parts than you would like to.