Adam and Eve were told that if they partook of the tree of knowledge of good and evil they would die, Genesis 2:17 and Moses 3:17. Adam and Eve did partake of the fruit, Genesis 3:6 and Moses 4:12-13. Which transgression brought about the The Fall, Moses 6:59.
The fall brought both spiritual and physical death into the world, Helaman 14:16-17. We would remain in our fallen state if it weren't for the Atonement of Christ, 1 Nephi 10:6, 2 Nephi 9:6-9, Alma 42:7, and D&C 29:40-42. All are cut off, but the Atonement covers little children and those who sin ignorantly, Mosiah 3:11-12 and Mosiah 3:16. All mankind are fallen and lost from God and become natural men, Alma 34:9. "For the natural man is an enemy to God, and has been from the fall of Adam," Mosiah 3:19. Since man has fallen they seek after the things that are "pleasing unto the carnal mind," Alma 30:53. They become carnal, sensual and devilish, Alma 42:10. Man became evil continually, Ether 3:2 All the people have become corrupt, Psalm 14:1-4.
The fall was a necessary part of the plan, 2 Nephi 2:22-25, 2 Nephi 9:6, Mormon 9:12, and Moses 5:9-12. There is opposition in all things, the forbidden fruit is in opposition to the tree of life, 2 Nephi 2:15-16. The devil provides opposition to our progress and in that sense is part of the plan, 2 Nephi 2:17-18 and D&C 29:36-39. The fall was necessary so that children could be born into the world, 2 Nephi 2:19-21. The fall separated us from God, but this mortal life was provided as "a probationary time, a time to repent and serve God," Alma 42:2-5 and Alma 42:11-15.
The way is prepared from the fall of Adam for the children of God to be saved, 2 Nephi 2:4. Even as death comes upon all of us because of the fall of Adam, the resurrection is is a free gift from Christ unto all mankind, 1 Corinthians 15:21 and Romans 5:12-18. Through Christ man are redeemed from the fall, D&C 20:20-21, D&C 93:38-39 and D&C 138:18-19. However, it takes repentance to be redeemed from the fall, D&C 29:43-45.
When Aaron was teaching the king over all the land of the Lamanites he started with the Fall of Adam to show the need for the atonement, Alma 22:12-14.
Moses wanted to atone for the sins of the children of Israel, but the Lord tells him in Exodus 32:33 that it doesn't work that way. There are a number of other scriptures that point out that one man can not atone for the sins of another. Only Christ can atone for other's sins. "The fathers shall not be put to death for the children,..." Deuteronomy 24:16 See also Ezekiel 18:20 and Alma 34:11-12.
Also see the Second Article of Faith.