Genesis 3:15 is known as the protoevangelium—the first gospel. The verse introduces two elements previously unknown in the Garden of Eden, elements which are the basis of Christianity—the curse on mankind because of Adam’s sin and God’s provision for a Savior from sin who would take the curse upon Himself.
Verse 14 makes it clear that God is speaking to the serpent who He curses to crawl on his belly and “eat dust” all his days. In verse 15, God switches from condemning the serpent to the one who inhabited it, Satan. He curses Satan to be forever at war against mankind, depicted as the seed or offspring of the woman. The woman in question is in a general sense Eve herself, all of whose offspring would forever be harassed by Satan and his minions. Sin enters the human race at this point, and the ravages of sin and its consequences reverberate down to us today. We inherit sin and the sin nature from Adam, and we suffer for it continually. The enmity—the hostility and hatred—of men and demons, between whom the warfare still continues, begins here. Evil angels and also wicked men are called serpents, and even a brood of vipers (Matthew 3:7), and they war against the people of God, the seed of the church, who are hated and persecuted by them, and so it has been ever since this affair in the Garden.
More specifically, the offspring of the woman refers to Jesus Christ, who was born of a woman. The “enmity” or hostility and hatred spoken of here is between Satan and Christ. The seed of the serpent, evil men and demonic forces, struck at the heel of the Savior when they entered Judas, the Pharisees, the rabble, and the Romans, who all conspired to condemn Jesus to be crucified. But His wound was not the final act. He rose the third day, having paid the price for the sin of all who would ever believe in Him. The ultimate victory was His, and He crushed the head of Satan, removing forever his rule over man. The power of Christ would destroy Satan and all his principalities and powers, confound all his schemes, and ruin all his works. The power of the cross would crush Satan’s whole empire, strip him of his authority (particularly his power over death), and his tyranny over the bodies and souls of men. All this was done by the incarnate Christ when He suffered and died for the souls of men (Hebrews 2:14–15). Because of what Jesus did on the cross, he “crushed” the devil’s head, defeating him forever. The protoevangelium shows us that God always had the plan of salvation in mind, and informed us of His plan as soon as sin entered the world.
 Got Questions Ministries. (2010). Got Questions? Bible Questions Answered. Bellingham, WA: Logos Bible Software.