How does the government determine 'essential' vs. 'non-essential' personnel?

According to Reuters, as man as one million government workers got to sleep in today after Congress failed to avoid a shutdown. But don't worry -- only "non-essential" personnel will be furloughed during the shutdown...

This got me wondering -- How do they decide which positions are "essential" and "non-essential"?

This is pretty much how I imagine that process: