Only primates have a period the way humans do. A heat is different. A period is when the egg has passed and is being "kicked out" A heat, on the other hand, is the time when a female is most fertile and she is trying to attract a mate.
They go through a cycle much like humans but in the case of animals this is the time that they breed. With humans one is not likely to get pregnant during their period (but you can). I am not sure if the male gender in animals goes through a period like human males do....LOL...and yes, you guys do have a period...it is just not physically like the females.