Medium means in the middle. It is often used in the context of temperature or size. For example, some shirt sizes are small, medium, and large.
Average means normal. If I have an average exam score, that means that my score is like everyone else's. Also, if I have a mother, a father, and a brother, you could say that I live in an average family.
Moderate means not extreme. If a day is neither hot nor cold, you could say that it has moderate temperatures. If my political views are neither on one side nor the other, you could say that I am politically moderate.