The woman mentioned in Genesis 3:15 is Eve. The serpent, addressed directly, is the animal that Satan used to deceive the woman. Some of the curse was directed at the animal (verse 14); at the same time, the curse of God falls upon Satan, who had taken the serpent’s form or body in Eden (cf. the dragon in Revelation 12:9)..... "THAT OLD SERPENT" (Rev. 20:2)
As part of the curse, enmity—mutual hatred and ill will—will exist between the woman and the serpent. Later, the same enmity will continue between the woman’s seed or offspring (mankind in general, since Eve is the “mother of all living,” Genesis 3:20) and the serpent’s seed. Their offspring will remain enemies throughout all generations. The serpent’s (metaphorical) offspring are demonic forces and also those people who follow the devil and accomplish his will.
Jesus called the Pharisees a “brood of vipers” in Matthew 12:34 and said they belonged to their “father, the devil” in John 8:44. In short, God says that Satan will always be the enemy of mankind. It follows that people who side with Satan will be at perpetual war with God’s elect and that we are engaged in a very real battle between good and evil (Ephesians 6:12).
Men normally have the "seed". In Jesus' case, he is the only person born from the "seed of the the woman", since there was no male contributor of DNA.