Eve is the name of the first woman on Earth. The Lord made her from one of Adam's ribs when he was asleep. She gave in to the snake's temptations to eat the forbidden fruit, and had Adam do so as well, despite that she knew full well God has forbidden it. As a punishment women were made subject to men and were forced to endure painful childbirth.
New Eve
The new Eve is the Blessed Virgin Mary, who by giving birth to Our Lord, gave mankind the "Fruit" Who would crush Satan's stronghold on man via original sin.