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). Female animals with overt menstruation are generally sexually active throughout their cycle. In comparison, females with covert menstruation are only ‘in heat’ mid-cycle.
Overt menstruation occurs in humans; most primates (including chimpanzees, organutans, gorillas & rhesus monkeys); some types of fruit bats; and elephant shrews. The average cycle length in orangutans and opossums is the closest to that of humans, 28 days, while the cycle for chimpanzees is 35 days. Menstrual bleeding in non-human primates is minimal.
Covert menstruation (or the estrous cycle) occurs in all other placental mammals (including 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. 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.
Egg-laying animals do not menstruate. This includes some reptiles, some amphibians, some fish, and all birds. Egg-laying animals do not need a uterus therefore they do not have a menstrual cycle.
See below for some photos of animals (other than humans) that have overt menstruation. From top-left, clockwise: gorilla mother and baby, chimpanzee, short-tailed fruit bat, elephant shrews, opossum, three opossums hanging from branch.