Law and Gospel is the basic difference between Christianity and all false religions.
The difference between the Law and the Gospel is that the Law tells us what we are to do, while the Gospel tells us what God does for us. From the Law comes only a knowledge of sin, death and condemnation, while the Gospel is a free gift of forgiveness, and every grace and blessing.
The basic error of all false religions is believing that, as bad as we may be, there must be at least something good deep within us by which we can reform ourselves by doing good works, and by 'becoming better' save ourselves.
By contrast, Christianity teaches "there is no one who does good," (Psalm 14:3), and we are "sinful at birth, sinful from the time my mother conceived me," (Psalm 51:5) so that we are "all objects of wrath" (Ephesians 2:3).
We cannot reform ourselves, because we are not just spiritually sick—we are dead! However, like Adam after he sinned, we are very good at covering up and hiding our sinfulness and pretending that it isn't so! And nothing separates us from God more totally than the self-righteousness which believes we have something good within us which ought to count for something before God!
The primary purpose of God's Law is to smash and condemn this self-righteousness. The Law does this by holding up before us all God says we are to be like: completely perfect in absolutely every way and always doing so with 100 percent pure motives and inclinations with perfect fear, love and trust of God and with all our heart, soul, mind and strength, (Mark 12:30-31), every minute of our lives. When we miss the mark even once, even in the slightest, the Law condemns us absolutely to eternal death and hell. Luther's explanations in the Small Catechism expose to us "how very low our poor nature has fallen, how we have become utterly corrupt." so that "through the Law we become conscious of sin" (Romans 3:20), and are put to death. The Law drives us to admit "I'm in deep trouble!"
But the Law is not God's ultimate word for us. He has another, more important one—the Gospel! In the Gospel everything is a gift. God gives us salvation by His own work for us, not ours for Him, which we do not have to earn. Jesus Christ alone lives the perfect life the law demands, and He bears the punishment for our sin upon the cross, and rises again to new life, giving us a share in this Life. "All have sinned and fall short of the glory of God, but are justified freely by His grace through the redemption which came by Jesus Christ" (Romans 3:24). The Gospel is the "power of God for salvation of everyone who believes" (Romans 1:16).
Only when the Church's ministry has both these words from God—properly distinguished—are sinners brought to repentance and faith in Christ and Christians kept in the faith!
God is like a doctor, and we are like a patient having a heart attack who thinks everything is fine, and if we can exercise a bit more, we can eventually get healthy again! To secure sinners, our chief Physician must preach only Law, and threaten us with God's judgment and death. We cannot make our sinful lives healthy again. The life into which we are born is unreformable and under God's sentence of death. There is nothing we can do about it!
However, when through the Law we become sinners alarmed over our sins, God then speaks His Gospel Word by which we receive a heart transplant—a new heart, a new life, the life of Christ Himself—freely by His grace! The Gospel has no threats or condemnations, only words of consolation!
Law and Gospel is not just one doctrine among many—it is the core of Christian faith, the key to a correct understanding of all aspects of Christian and Lutheran teaching. It is the framework by which you read the Scriptures and listen to sermons. And it is the shape of your life as a Christian, as you "by daily contrition and repentance [from the Law] are drowned and die with all sins and evil desires, and that a new man [by the Gospel] should daily emerge and arise to live before God in righteousness and purity forever" (Small Catechism, Baptism).