I don’t know what this is my nipples are puffy and I don’t like it - how can I get them fixed? (Photos)

Hello, Thank you for your question regarding puffy nipples and the plastic surgery treatment of puffy nipples. It appears from your pictures that you have glandular tissue underneath your nipple areolar complex. For definitive treatment and correction of puffy nipples, I would suggest VASER assisted liposuction to remove this sub-areolar glandular tissue. VASER liposuction to remove male breast gland tissue forgynecomastia is usually very successful. Please seek an experienced, board-certified plastic surgeon for the treatment of your puffy nipple and gynecomastia. Again, thank you for your questions regarding puffy nipples and the treatment for gynecomastia. Before you make any decisions regarding surgical treatment of puffy nipples, I wouldsuggest you seek consultation with an expert. Please seek an experienced, boardcertified, plastic surgeon. Travel out of state if you must to find such gynecomastia correction expert. If you have any more questions regarding gynecomastia surgery or plastic surgery in general, please contact my office.Sincerely, Dr. Katzen. (Certified by the American Board of Plastic Surgery and member of the American Society Plastic Surgeons, International Society of Aesthetic Plastic  Surgery, American Academy of Cosmetic Surgery, and Fellow of American College of Surgeons and American Board for Metabolic and Bariatric Surgery)

There are a lot of factors to consider when it comes to puffy nipples. An examination would have to been done to find out what best fits you.
This is a common procedure that most people don’t even know is an option. Surgeons can improve puffy nipples with a wide variety of surgical and nonsurgical simple office procedures under local anesthesia.
This can typically be done in the office under local anesthesia. Usually there are no drains, no outside stitches, and patients can go out in public the same day. Most of the initial swelling is from the numbing medicine and your body absorbs it within the first 4-6 hours. Obviously the body heals with swelling and inflammation and usually that maxes out at two days and then starts getting better. Most of these patients take no pain medicine and usually feel very comfortable going out in public within a few days.  I usually call my patients the night of surgery and check on them the next day. If everything is fine the next day, I then let them shower and they do not need a dressing over their incisions. They can usually start using make up after a few days.  At the two week check up, I usually have people start working out and then after six weeks they can do whatever they want. 
Additionally with recovery, there're many things we do to expedite healing such as nerve blocks during surgery, oral supplements, skin care and scar therapy recommendations, lymphatic massage sessions that are included in our pricing.
I strongly feel that it's very important to have an honest conversation with your surgeon about your particular situation, lifestyle, work environment and specifically which procedures you're having done so that you both have the most honest answer about your recovery and are realistic with expectations and results.
I would recommend that you see a board-certified plastic surgeon in your area that is a member of both ASPS & ASAPS and performs a lot of breast and nipple surgery.
I hope this helps and wish you all the best in your surgery and event. If it does, please take a moment to up vote this answer on the “respond” button on the top right.  

Thank you for your question!  Based on the photos that you have provided, you will benefit from a male breast reduction, also know as Gynecomastia.  Gynecomastia is caused by the swelling of glands beneath your areola/nipple.  The surgery will be performed by excising the glands and eliminating the puffiness that you are experiencing.  In some cases, liposuction is performed in conjunction with this surgery to improve the overall contour of the chest.  I recommend that you see a board certified plastic surgeon to obtain your customized treatment plan.