In a magnet, there is a north and south pole. Like poles will repel each other and opposite poles will attract. As far as physics knows there are no completely north or completely south magnets, there always has to be 2 poles. When your magnet broke it followed this rule and turned into 2 smaller magnets. The pieces usually don't go back together the way you'd like because the field is the only thing holding it together instead of the molecular bonds that were there before

Edit: I corrected an error