Feedback describes any system where the output of that system is returned to its input.  In public address systems or studio monitoring this term (acoustic feedback) has a pejorative connotation (see below).  In many electronic and digital audio systems feedback is used constructively: it may be found in many circuits and algorithms.
In situations where the monitoring system is in the same room as the recording care should be taken to avoid the unpleasant and potentially dangerous phenomena known as acoustic feedback. 
Feedback occurs when the sound picked up by the microphone is amplified and sent to the loudspeakers and back into the room where it is again picked up by the microphone, further amplified, and so on, leading quickly to an infinite amplification loop and a loud squealing sound. Feedback is loud, annoying, and potentially dangerous to ears and loudspeakers. 
When recording and monitoring the signal being picked up by the microphone be sure not to have the loudspeaker level turned up. Either use visual confirmation by watching the meters on the mixer or recorder, or use headphones to listen to the sound as it is recorded.  Conversely, when playing back the recorded sound, make sure the mixer channel being used for the microphone is turned down.