Keep in mind that polytheism is not limited to the religions of ancient Greece and Rome. In reality, a large number of religions, including indigenous religions, are polytheistic. (We will discuss the issue of polytheism in Hinduism below). While the pantheon of polytheistic religions can include hundreds or even thousands of gods, two deities that are almost always present are the Sky God and Mother Goddess.
The Sky God
As the name implies, the Sky God is generally associated with the sky and the life-giving rain that comes from the sky. In some cases (especially in traditional African religions), the Sky God may be a powerful creator, who has essentially withdrawn from the day to day events of gods and human beings. In other cases, the God functions as the ruler of the lesser gods. In this monarchical polytheism, the Sky God is capable of impacting and intervening in the affairs of both divine and mortal beings. An obvious example of a Sky God is the ancient Greek God, Zeus According to Greek mythology, Zeus dwells in the sky and uses thunder and lightning as his weapons. In addition, he is the most powerful of the Greek gods and rules over both human beings and the gods.
In almost every polytheistic religion, there is a Mother Goddess who is the most prominent of the goddesses. She is associated with fertility and the change of seasons. In many cases, her death, veiling, or seasonal journey to the underworld is the cause of the death of vegetation each year. When she returns to life, unveils her face, or leaves the underworld, the vegetation and new life appear on the earth.
One prominent example of the Mother Goddess is the ancient Akkadian Goddess, Ishtar. Ishtar was the goddess of sexuality and fertility. According to Sumerian mythology, Ishtar journeyed to the underworld in a vain attempt to bring back her lover Tammuz. Although she fails to bring Tammuz back to the land of the living, she makes arrangements to spend six months out of each year with him in the underworld. These six months correspond to fall and winter. Her return from the underworld corresponds to spring and summer. (As an extra bit of information, this myth is referred to in the Hebrew scriptures in the book of the prophet Ezekiel).
The Ishtar myth has its counterpart in the story of the Egyptian goddess Isis, the Greek goddess, Persephone, and the Anglo-Saxon goddess Ostara. As another example of how Christianity incorporated pagan beliefs and practices, note that the day of Easter takes its name from the goddess Ostara.
Feminine Aspects of the Divine
As you text notes, not all goddesses are Mother Goddesses any more than all gods are Sky Gods. In fact, in all cultures, goddesses can play a variety of roles including warriors, destroyers, healers, etc. In Hinduism, sakti is the name given to the feminine sacred energy. This energy is manifest in a variety of goddesses with a variety of different roles. In many cases, the goddess is a companion to a god. In other cases, the goddess may more or less stand alone in fulfilling her role.