Eve sinned first in the Garden of Eden, but sin entered the world through Adam (Romans 5:12).
"The LORD God commanded the man, saying, “From any tree of the garden you may eat freely; 17 but from the tree of the knowledge of good and evil you shall not eat, for in the day that you eat from it you will surely die.” ...3:1 "Now the serpent was more crafty than any beast of the field which the LORD God had made. And he said to the woman, “Indeed, has God said, ‘You shall not eat from any tree of the garden’?” 2 The woman said to the serpent, “From the fruit of the trees of the garden we may eat; 3 but from the fruit of the tree which is in the middle of the garden, God has said, ‘You shall not eat from it or touch it, or you will die.’ ” 4 The serpent said to the woman, “You surely will not die! 5 “For God knows that in the day you eat from it your eyes will be opened, and you will be like God, knowing good and evil.” 6 When the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was desirable to make one wise, she took from its fruit and ate; and she gave also to her husband with her, and he ate. 7 Then the eyes of both of them were opened, and they knew that they were naked; and they sewed fig leaves together and made themselves loin coverings," (Genesis 2:16-17; 3:1-7).
We see from the text that God told Adam he could eat from any tree in the garden except for the tree of the knowledge of good and evil. Afterwards, God made Eve out of Adam's side. Though the text does not say that Adam specifically told Eve what God had said, we can conclude that he did. Eve knew not to eat of the fruit (Gen. 3:2). Then the serpent, who was the devil, deceived Eve and she "...took from its fruit and ate and she gave also to her husband with her, and he ate."
So, she ate of the fruit first and she gave it to her husband who then ate with her. So, she sinned first, and Adam sinned second.
Adam was with Eve
Some Christians assert that Adam sinned first because he did not prevent Eve from eating the fruit since he was "with her" (Genesis 3:6). But, when we ask what that phrase "with her" means, we cannot logically determine that it means he was standing right next to her and was fully aware what was going on. The text simply says that he was there with her. But, what does it mean? Does it mean he was with her in the garden far away? Does it mean that he was 50 feet away? Without knowing, we cannot say that he was standing next to her during this dialogue Eve had with the devil. So, it cannot be maintained that Adam actually committed the first sin by permitting Eve to eat of the fruit.
As Eve was tempted by the devil, Adam was tempted by Eve.
Sin entered the world through Adam
We know that sin entered the world through Adam because Paul says so in Romans 5:12, "Therefore, just as through one man sin entered into the world, and death through sin, and so death spread to all men, because all sinned." But if Eve sinned first, how was it that sin entered the world through him and not her? The answer lies in the doctrine of federal headship which says that the male represents the descendants, not the female.
Adam was the one who represented mankind, not Eve. This concept of federal headship is taught in the Bible in several places but if you want to know more about it, please see the article Federal Headship.