A breast abscess is a pocket of pus.  The first photo shows a breast abscess about to drain on its own. In the breast, an abscess arises often when treatment of mastitis is delayed and/or the mother is given inappropriate treatment.  The bacterium which almost always  causes an abscess Staphylococcus aureus which is resistant to drugs such as amoxicillin and erythromycin; yet the last two mothers who emailed me about an abscess were being treated with amoxicillin. The antibiotics of choice for mastitis include, cephalexin, penicillin derivatives such as cloxacillin or dicloxacillin (but not plain penicillin), cephalosporins such as cephalexin and others such as clindamycin and if there is concern about methicillin resistant staphylococcus aureus (MRSA), then cotrimoxazole (Bactrim or Septra) is one possibility.  Intravenous vancomycin is another.  None of these antibiotics require a mother to interrupt breastfeeding.

Another reason that mastitis may turn into an abscess is that the mother is told not to feed on the side affected by mastitis.  This is incorrect.  It has been a surgical principle for years that one needs to drain an area of infection.

An abscess needs to be drained; antibiotics alone will not treat an abscess.  Surgeons have, in the past, almost always did surgery on the breast (incision and drainage) and almost always, I hear, tell the mother not to feed on the affected side, though sometimes telling her not to breastfeed at all, not even on the other side.
There is a better way.  See the second attached photo.  Here is a link to an article published in Radiology in 2004.  And here is the link to the chapter in my book that discusses sore breasts and our experience with this treatment as outlined in the article.  Note that the mother in the two photo can, and should, continue breastfeeding on the side where the catheter (which drains the abscess) has been placed.
In our experience of over 100 breastfeeding mothers with breast abscess over 12 years or longer, we have had only one recurrence which was treated in the same way and then was cured, and only one case where we were worried that a fistula had formed (when a wound does not heal and continues to leak fluid, it is called a fistula), but in fact it healed without the mother interrupting breastfeeding on that side.