What is wrong with my nipples? I'm not sure if I have puffy nipples or something else but I hate the way they look. (photo)

Pretty much everything was said in my title, but you really should
understand that you are not even close to either end of the bell-shaped curve
of "normal nipples," you are solidly right in the center.
I'm not sure where you get your idea that small(er), non-pouty, and non-puffy
are somehow "beautiful" and different from your nipples, but your
anatomy is NORMAL and actually very attractive. Like them while you can,
because, trust me, they will change with age, pregnancy, breast feeding, and
weight changes.
I am certain you can find some surgeon somewhere who would be willing to
produce some sort of surgical change in your nipples, but I would wager that
the potential loss of ability to breast feed or change in sensation would not
be worth the very minimal appearance change that could be produced, even if the
scarring was truly minimal.
Please, do not be concerned that your breasts are unattractive--I've seen
thousands of breasts, and most of them would not be in a centerfold. Yours are
completely normal, pretty, and very attractive. Sometimes a woman's feeling bad
about a certain aspect of their anatomy comes from an unkind tease from another
girl in a locker room (frenemy), or a total jerk of a guy on a date (where
perhaps intimacy has preceded the usual kindness that love would engender).
Neither should be taken seriously, though words can stick in our minds. Even
jealous mothers can sometimes be unintentionally cruel. Regardless of what
caused this concern, I would strive to get you to understand that no surgery is
necessary or even advisable. For examples of women with all kinds of nipples, click on the web reference link below. Best wishes! Dr. Tholen

Your nipples do not look that unusual to me. However, you can discuss them with a plastic surgeon and maybe a simple nipple reduction may make them better for you. Good luck.

I am sorry to hear about your concerns. Your best bet would be to meet with board certified plastic surgeons to discuss your concerns. Try to be as specific as you can be about what you "hate" about the nipple anatomy. It is likely that working together you will come up with a plan to achieve your goals. Best wishes.