Yes, many animals do menstruate, but only a handful menstruate overtly like humans do (where there is blood flow from the uterus through the vagina). Other animals menstruate covertly (by simply reabsorbing the uterine lining into the body).
Overt menstruation occurs in humans, chimpanzees, organutans, gorillas & rhesus monkeys . Chimpanzees start their cycle when they are about 10 years old, and their cycles last for an average of 35 days (a bit longer than that of humans). Menstrual bleeding is minimal.
Covert menstruation (or the estrous cycle) occurs in most placental mammals (dogs, cats, horses, elephants, rats, pigs). Dogs have their cycle twice a year, cats have it about once a month, horses have it every three weeks, and rats go through their cycle rapidly – every 5 days (no wonder they’re so fertile!). Some of these animals (eg. dogs) display a bloodily discharge from the vagina due to declining estrogen levels. This is different from menstruation because it occurs when the animal is ‘in heat’ during mid-cycle (at the time of ovulation) and the blood comes from the vaginal walls, not the uterus.
Female animals with overt menstruation (like humans & chimpanzees) are generally sexually active throughout their cycle. In comparison, females with covert menstruation (like dogs & cats) are only ‘in heat’ mid-cycle.