Automatic mixers make life easy for churches or schools who need to hook up several different audio sources, but don't have or can not use a dedicated sound engineer on a regular basis.
To put it simply, automatic audio mixers sense a signal coming in through an input source, and pass that signal on to an amplifier or whatever the next signal processing unti is in the audio chain. Typically a church will use an automatic mixer to manage the several microphones that are on the altar. When the automatic mixer sees that audio is coming into it, it turns that particular channel on and sometimes, but not always, closes off or lowers the volume on the channels where it does not detect an incoming audio signal. There are many ways to program an automatic audio mixer, so I am just giving you one specific example.
In closing, an automatic mixer opens an audio pathway on channels where it detects an audio signal, and closes the audio pathway on channels where it does not detect any audio signal.
Automatic mixers can vary widely in price. The price generally swings based on how many channels the mixer can manage and how many complex tasks it can perform on the audio signal.