When Christians or others speak of Adam as the first sinner, this comes from the Apostle Paul where he states,
Therefore, just as through one man sin entered into the world, and death through sin, and thus death spread to all men, because all sinned. (Romans 5:12)
It means that sin entered the world through Adam — that he is the one credited with sin's entrance and hence the subsequent entrance of death and suffering and the need for a Savior — a last Adam (1 Corinthians 15:45). When we look back at Genesis, it is true that Satan rebelled, and also Eve sinned, prior to Adam's disobedience.
The Sin of Eve
There were several things that Eve did wrong prior to eating the fruit. When the serpent (who was speaking the words of Satan) asked in Genesis 3:1, "Has God indeed said, 'You shall not eat of every tree of the garden'?" her response was less than perfect:
And the woman [Eve] said to the serpent, "We may eat the fruit of the trees of the garden; but of the fruit of the tree which is in the midst of the garden, God has said, 'You shall not eat it, nor shall you touch it, lest you die.'" (Genesis 3:2–3; emphasis added)
Compare this to what God had commanded in Genesis 2:16–17:
And the Lord God commanded the man, saying, "Of every tree of the garden you may freely eat; but of the tree of the knowledge of good and evil you shall not eat, for in the day that you eat of it you shall surely die."
[Read the rest of the article at Answers in Genesis.]