What causes my chain to derail at the rear when I accelerate? I recently bought a beat-up old mountain bike for commuting. I've tried to clean and adjust the derailleurs and everything seems to work well enough in my garage. But when I put heavy weight on the pedals (say, starting across an intersection after a stop light) the rear gear momentarily derails with a crash (or do you say the chain derails?), I flounder for a second, there's another ugly clashing sound, and then things go back to normal. Is this something that I can adjust my way out of or more likely an indication of a worn-out part?

It sounds like the combination of your chain and cassette / sprocket are at the end of their lives. Examine them for signs of wear.

To check a cassette for wear, look for pointy teeth. A non-worn cassette will have teeth with a fairly flat ends.

A worn cassette will have teeth that have become pointier.

To check your chain for wear, ideally you'd use a chain wear tool to check how much your bushings have been worn away. If you don't have one of these tools, you can kindof tell when a chain is worn out because it will feel like the bushings are a bit loose.

Alternately, a drivetrain generally wears out at around 10000 kms (~6000 miles). You can rack this amount up pretty quickly as a regular commuter.