The basic symptom I was having is that sound was not working in Ubuntu 14.04.
However, I had recently used the front headphone jack, and after unplugging the headphones, I noticed in the sound settings control panel that the headphones output device briefly disappears, then immediately re-appears. This leaves the computer in a state where it doesn't play any audio.
In other words, it's stuck in headphone mode. This may be a hardware problem, but given the solution I found (below), I kind of doubt it.
Best Answer
To fix this problem, I did the following:
This caused the headphone output to properly disappear, leaving the "Analog Output" in a working state.