"...for through the Law comes the knowledge of sin," (Rom. 3:20).
In the Old Testament God gave the Law through Moses. It is the commands and precepts that govern human conduct. In the New Testament God gave the Gospel through Jesus. It is the message of salvation by grace through the sacrificial death and physical resurrection of Jesus, for our sins.
The Law is the do's and don't's of moral behavior. It consists of the 10 commandments (Exodus 20), rules for social life (Exodus 21:1-23:33), and rules for the worship of God (Exodus 25:1-31:18). It was a covenant of works between God and man and was (and is) unable to deliver us into eternal fellowship with the Lord. The Law is a difficult taskmaster because it requires that we maintain a perfect standard of moral behavior. And then when we fail, the Law condemns us to death. Works do not earn us salvation or play any part of it. The Bible says that a man is justified by faith apart from the works of the Law (Rom. 3:28).
The Gospel, on the other hand, is the good news of the death, burial, and resurrection of Jesus for our sins (1 Cor. 15:1-4). It is the message of what God has done for us, our deliverance from sin and the punishment of the Law. "Law and Gospel" are also part of our foundation, and a good understanding of their relationship will greatly help your witnessing. How? If you understand that the Law of God is a standard of perfection, that it reveals sin, that we are unable to keep from breaking it, and that the Gospel frees us from the need to keep the law perfectly in order to obtain forgiveness of sins, you will then be better able to communicate the message of salvation to the unsaved.
The Law is different from the Gospel
Most Christians already have a basic understanding of the difference between Law and Gospel; they just don't know they do. For example, "You are a sinner (Law). You need Jesus as your Savior (Gospel)." The Old Testament (Law) came before the New Testament (Gospel). The Law shows us what we are guilty of and the Gospel delivers us by grace. First we must know we are guilty (Law) before we recognize our need to ask for forgiveness (Gospel).
The Law kills. The Gospel makes alive. When Moses came down from the mount after receiving the Law and saw that the Israelites had fallen into idolatry, he threw the tablets of the Law down to them and 3000 people died (Exodus 32:28). Later, when Peter preached the Gospel, 3000 people were saved (Acts 2:41).
With a better understanding of the Law, it will be easier for you to explain sin. Without the Law, sin cannot be known; Romans 3:20 says, "...through the Law comes the knowledge of sin." (See also Rom. 7:7). If sin is not known, then the need for Jesus is not felt. This is why you mention the Law to those with whom you witness. How? By asking them if they have ever sinned. Tell them that lying, cheating, stealing, lusting, not honoring God, etc. is sin. Everyone is guilty somewhere (Rom. 3:23), so everyone needs to be delivered. Everyone needs the Gospel.
The Law is peculiar. It says "be holy," but shows us we are not. It says "do not lie," but shows where we do. It says "honor the Lord your God," yet shows us where we fail. Since none of us can keep the whole Law, we are all under condemnation. There is no way out. What can we do? Nothing! That is why "the Law has become our tutor to lead us to Christ, that we may be justified by faith" (Gal. 3:24). The Law guides us to Him. How does it do that? By showing us that the attempt to keep the Law (our works) is insufficient to gain eternal life and that the Gospel of grace is the only way to God.
In other words, you must help the person realize that they are not good enough to merit God's favor. People tend to think that because they are sincere or "not that bad" they are going to be with God when they die. But the Bible reveals that "sincerity" and being "not that bad" are not good enough. God requires perfection.
Salvation is of God.
That is why salvation belongs to the Lord (Psalm 3:8), by faith and not by works (Rom. 4:5). That is why it is a free gift of God (Rom. 6:23), through grace (Eph. 2:8-9). That is why God became man (John 1:1,14) and fulfilled the Law: "For what the Law could not do, weak as it was through the flesh, God did; sending His own Son in the likeness of sinful flesh and as an offering for sin, He condemned sin in the flesh" (Rom. 8:3). And also, "For if a law had been given which was able to impart life, then righteousness would indeed have been based on Law" (Gal. 3:21); "For by grace through faith you have been saved, not by works..." (Eph. 2:8). And, "...but to the one who does not work, but believes in Him who justifies the ungodly, his faith is reckoned as righteousness" (Rom. 4:5).
In presenting the Gospel, you show how the requirements of keeping the Law perfectly is removed. Say something like, The Bible says that if you break just one command of God, you are condemned, (James 2:10-11). I often add, "Sin can be forgiven but the effects continue. The effect of your sin is death. Your sin is an offense to the Law-giver, God. But Jesus, who is God in flesh, bore our sins on the cross and died with them. If you want your sins forgiven, then you need to come to Christ and ask Him to forgive you. He will."