Q:

Receive an email when a UNIX process fails

There is a process running on our server that fails every once in a while.   I'd like to receive some sort of email notification whenever that happens.   Is there a simple way to set that up? 

A:

The simplest way to do it would probably be this one-liner (inspired by SpliFF's answer):
failing_process || mail -s 'Process failed' localpart@example.com

For a more sophisticated monitoring of your system (and not only a single process), take a look at Nagios, monit and all the other well-known monitoring tools.  They usually also support notifications by other means than just by email. 
For a single system, monit is probably the most suitable solution.