This is one of those "Let my life serve as an example" sort of posts. :D
I'm usually pretty good about keeping an eye on chain wear so that I don't have to do a wholesale drivetrain "upgrade" but failed miserably this time. I hadn't messed with the chain on the fat fixie for wayyyyyy tooo long. I went to put a new chain on last night and when I went for a shakedown ride, all I heard was crunching and grinding out of a perfectly clean and partially new drivetrain. What should have been a $20 fix ended up costing me $150.
Background: As your ride, your chain stretches and wears. If you don't clean your chain often, that process happens quicker. If you ride with a chain that is stretched and worn, the cogs (gears in the back) and the chainrings (gears in the front) wear to match the longer, worn out chain. When you finally do get around to replacing your chain, the bike will no longer pedal or shift correctly. Any time you put pressure on the pedals, the gears will pop and grind.
The "solution": If you've waited too long the only solution is to change the chain, chainrings and cogs all at once. That gets expensive. One reason you see seemingly nice, lower-end bikes on Craigslist at a good deal is that someone paid $500 for a new bike two years ago, ignored the chain and just heard from their mechanic that is going to cost $300 to replace chain, cogs and chainrings. Buyer beware!
How do I avoid this?: There are many companies that make tools that measure how much your chain has worn. They're simple to use and quite reliable. Use them frequently... especially if you ride when it is wet or don't clean your chain often. When the gauge says that your chain should be replaced... or is getting close to needing to be replaced, buy a new chain and do it!!! The fix is easy to do. It is a great job for someone wanting to start doing their own repairs.
Chain wear gauge: http://www.parktool.com/product/chain-wear-indicator-CC-3-2
Chain wear video: http://www.youtube.com/watch?v=6pUgHM8HU2Q
Everything you ever wanted to know about chains and much, much more from Sheldon Brown (MHRIP): http://sheldonbrown.com/chains.html
Hope that is somewhat helpful.