How do I get rid of my puffy nipples? Not man boobs. (photos)

Hi and thanks for your question. I would suggest a reduction of the areolar skin  (the areola is the dark circle, the nipple is that which is in the center and erects when its stimulated). Believe it or not, I also feel that although you don't have the typical man boob or " bitch tit", you do have some prominence of the breast tissue which is causing your nipple and areola to "pop" out. Therefore I would also perform a removal of this small amount of breast tissue to give you a better contour.Always seek out a surgeon certified by the American Board of
Plastic Surgery with years of experience in performing the procedure(s) that
you are contemplating. Good luck

Either directed surgical excision or liposuction of some type are the better options for your issues... 

Your nipples are normal, but your areolae (pigmented tissue around the nipples) are puffy.  This is common in gynecomastia (enlargement of the male breast).  Although the remainder of your chest appears normal, this is approached in the same way as any other male breast reduction.Surgery involves an outpatient procedure under general anesthesia.  A small incision is made at the junction of the pigmented and non-pigmented skin beneath the areola, and excess tissue is removed beneath the areola, leading to a flatter appearance.  An additional benefit may be a decrease in the diameter of the areola following this procedure.