When someone mentions "With the Mother" for Apple Cider Vinegar they're referring to the bacteria that is added to the Apple Cider Vinegar.
The Mother of the Apple Cider Vinegar is the bacteria added to the Apple Cider Vinegar and This bacteria is what is known as the "Mother", because it is the catalyst that gives rise to the Vinegar.
The "mother" of the Apple Cider Vinegar helps ailments better when taken with the Apple Cider Vinegar than it does without.
However Apple Cider Vinegar still works great without drinking the "Mother" portion of the Apple Cider Vinegar but it works much better when drank with it.