No, you are born with all of the parts you are ever going to have. What happens is that the parts you have, the ovaries mature and start producing eggs, the eggs if not fertilized come down the fallopian tubes and into the uterus where they attach to the lining. The the uterus sloughs the lining off once a month. That is why women have a monthly period, it is a cyclic refreshing of the uterine lining.
Actually, the answer is no. There arent any new body parts or new growths happening but what is happening during your menstruation is your body is ridding itself of the lining of the uterus. A hormone called estrogen thickens the lining of the uterus and ovulation begins. Ovulation produces an egg if the egg remains unfertilized it will be eliminated naturally through the vagina along with the thickened lining of the uterus.