Having a menstrual cycle is very rare in nature. A period is a phase in the ovarian cycle where the endometrium - the lining of the uterus - is shed. It will then regenerate each cycle, because for whatever reason, embryos only implant in fresh endometrium.
Only a handful of animals shed the endometrium and have a menstrual cycle. For all other mammals, they get rid of old endometrium in a different way: they reabsorb it if they don't get pregnant.