“your seed and her seed.” The Hebrew word for “seed” can be singular or plural, like “seed” in English. That is why some translations say “offspring,” indicating more than one. Also, although used technically the female has an egg while the male has a sperm or “seed,” the Hebrew word “seed” was used for offspring in general, as it is here in Genesis 3:15 (cp. Gen. 4:25; 9:9; 24:60; 46:6; Ruth 4:12; Esth. 9:27). The singular “seed” (offspring) of the woman, which was of primary concern to God at this time, was Christ, who would ultimately defeat and destroy the Devil (cp. what God said to Abraham; Gen. 12:7; Gal. 3:16), although the general nature of the prophecy would have included those among Eve’s “seed” (offspring) who were godly people and who throughout history have opposed the Devil and his minions.
The “seed” (offspring) of the serpent is all his children. For more on the children of the Devil, see commentary on Matthew 12:31.
“I will put hostility.” Once the Devil knew that he would be destroyed by one of the offspring of the woman, he started an aggressive campaign against them. At first it was against all people, and resulted in God rescuing humankind by Noah’s flood. As the people who would bring forth the Messiah narrowed, the intensity against the progenitors of the “seed” increased; thus, for example, came the aggressive attacks on the Jews.