note: all information in this writeup is with regards to human anatomy
In case anyone was not entirely clear about it, the nipple is the thing that sticks out from the breast. The areola is the coloured area around the nipple.
Also known as the mammary papilla, the nipple is the outlet for 15-20 lactiferous ducts which are arranged cylindrically around the tip of this projection of the breast.
Small non-striated muscle cells (myocytes) arranged cylindrically within the nipple are responsible for the nipple becoming erect when they are stimulated (for example, by suckling).
Embryologically, nipples develop along the 'milk lines' which extend from mid-clavicle down to the pubis on either side in humans. Most people develop two (one on each side) but some have supernumerary nipples. Occasionally, the extra nipples have lactiferous glands attached.
The physiological purpose of nipples in humans is to express milk during lactation. Sometimes, babies (male or female) are born expressing milk. This is called "witches' milk", is caused by maternal oestrogens acting on the baby and is quite normal. Witches' milk disappears after several days.