1.-When part of something breaks off or when you break it off, this part of the main object is removed by being broken.
Peter broke off another piece of bread and chewed at it.
I broke a branch off and stabbed at the ground with it.
A little bit has broken off the left hand corner.
2.- If you break off when you are doing or saying something, you suddenly stop doing or saying it.
He would break off the rehersal...I though - he broke off, then smiled"Sorry, not my business".
3.- If you break off a relationship or agreement with someone, you end it.
Men seem to be more skilled at breaking off relationships than women.
I've broken off my engagement.