The "man" was a figure or type of the Father, or God. There was already enmity between God and Satan, so this would have been nothing new. But the "woman" represented "the mother of all living" (see Genesis 3:20)--or of all mankind.
In the case of Jesus, the Seed that was to come, his Father was God, whereas Mary, his mother, was human. It was between humanity and Satan that God was to put enmity, or, more specifically, between the children of God and Satan.
As the Son of God, Jesus fulfilled this promise perfectly; and Adam and Eve understood the prophecy for what it was, which is why Eve mistakenly assumed her firstborn must be that Seed which had been promised, and she said: "I have gotten a man from the LORD" (Genesis 4:1, KJV). Unfortunately, this promise of a Seed was not to be directly fulfilled during her lifetime.
The typology of the man and the woman is representative of the relationship we have with God. These types are there to teach us precious lessons about salvation. They are not there to create gender conflicts--in a sense, it really has nothing to do with male versus female at all. The male and female representation is there to illustrate the kind of relationship God desires to have with His people. And the Seed of the woman is a type of Christ.
As Paul says:
Now to Abraham and his seed were the promises made. He saith not, And
to seeds, as of many; but as of one, And to thy seed, which is Christ.
(Galatians 3:16, KJV)
This is paralleled in Revelation.
Therefore rejoice, ye heavens, and ye that dwell in them. Woe to the
inhabiters of the earth and of the sea! for the devil is come down
unto you, having great wrath, because he knoweth that he hath but a
short time. And when the dragon saw that he was cast unto the earth,
he persecuted the woman which brought forth the man child.
(Revelation 12:12-13, KJV)
The woman represents Christ's church--his people. The "man child" is the Seed, the Promised One, the Messiah which was to bring us the promise of salvation.
And the dragon was wroth with the woman, and went to make war with the
remnant of her seed, which keep the commandments of God, and have the
testimony of Jesus Christ. (Revelation 12:17, KJV)
Understanding that the "enmity" must exist between this Seed and Satan, it helps us to understand more fully the intent of Genesis 3:15. Satan was told that the Seed was to "bruise thy head, and thou shalt bruise his heel."
When Jesus died on the cross, it was temporary. He was raised to life within three days. But when, in the end of this conflict, Satan is destroyed, he will not rise again--his head, not merely his heel, will have been "bruised."
The "heel" being bruised also seems interesting. Vipers in the Bible are depicted as biting at the heels.
Dan shall be a serpent by the way, an adder in the path, that biteth
the horse heels, so that his rider shall fall backward. (Genesis
The primary significance of the curse on the serpent is that the future final destruction of Satan is predicted. The Seed of the woman will bruise his head, and the serpent will only be able to bruise the heel of the woman's Seed. Inasmuch as God's people in general may be thought to have a part in the sufferings of the Seed (see Matthew 10:22-24) as children of God, Satan will not have the power to destroy them, only to hurt them during their journey--and in the end, it is Satan who will be destroyed.