Nipples get hard in the cold in order to preserve heat. If a nipple is exposed to the cold the automatically contracts, and this in turn causes the nipple to become smaller, wrinkled and hard. The reason this happens is to reduce the amount of surface area on the nipple that is exposed to the cold. As a result of this the body losses less heat and the amount of energy required sustain homeostasis (keeping normal body temperature) is decreased. Decreasing surface area of the skin is the ultimate reason why nipples get hard when it is cold.
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Nipples are made of erectile tissue and respond to stimulation. Sexual pleasure is secondary to having a surface for a breast feeding baby to latch on to.
To warm them up. If they contract, then that produces heat. They can also contract to give milk. Though in men, this is basically useless.
Sneezing causes a quick jump in blood flow. Because the nipples are an erectile organ, such a blood rush causes them to become temporarily erect.