Mix-minus is a particular setup for a mixing console, such that the output to a specific device contains everything except the input from that device.   The advantage of a mix-minus is the elimination of echoes and feedback.

A mix-minus signal is sometimes also known as a 'clean feed' or a 'Select Audio Return' (SAR).

How is it used? 

Back in my broadcasting days, I did some remote broadcasts from county fairs.  "I'm broadcast live from the  ___ County Fair..."  You get the idea.  During these broadcasts, I had a monitor so I could hear what was broadcast over the air.  This way, I knew when to talk and when to stop talking.  However, I never heard my own voice.  If I had, due to the communications, I would have heard a slight delay which would have been super distracting.
 

You might have a situation where you have an outdoor event that is broadcasting into a sanctuary, such as a pool baptism or other outdoor activity.  You might have a youth rally with events indoors and outdoors where you are showing outdoor activities within the church.  Using a mix-minus setup, you can get the sound you need into the church without the issues that would be caused by repeating the input signal back out to the source.