Eve ate the fruit first, and then gave some to Adam. According to Genesis 3:6, "She took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat."
The fruit eaten by Adam and Eve is not specifically identified in the Bible, but it is often referred to as the forbidden fruit or the fruit of knowledge. It is identified by God in Genesis 3:3 only as the fruit from the tree in middle of the garden, which was forbidden to Adam and Eve to eat, or else they would die. The serpent, when tempting Eve, said that eating the fruit would open her eyes and she would know of good and evil.