The Law that Jesus came not to abolish is the Law of the Gospel, which is the true Law of God. But Jesus Christ abolished many commandments of the Old Testament because they were not God's true Law. The Gospel says so:
"It was also said, 'Whoever divorces his wife must give her a certificate of divorce. Nevertheless I tell you that anyone who divorces his wife, except for the cause of prostitution, makes her commit adultery, and whoever marries a divorced, commits adultery" (Matthew 5:...)
The jews of that time wrote in the law of the Old Testament that could separated from women for many reasons when they tired of them, but Jesus abolished these commandments. These commandments were written into the Old Testament and Jesus Christ abolished them.
"Again you have heard that it was said by them of old not swear falsely, but shall perform your oaths to the Lord. Nevertheless I tell you, Do not swear at all, either by heaven, for it is the throne of God, or by the earth, for is the footstool of his feet, or by Jerusalem, for it is the city of the great King. neither shalt thou swear by thy head, because you can not do a single white or black hair. But let your ' Yes, yes, no, no, because what is more than these cometh of evil" (Matthew 5:...)
The Jews were accustomed to swear in many cases, and so wrote it in the laws of the Old Testament... The Lord abolished these precepts of the oaths.
The law of retaliation ("Eye for eye, and tooth for tooth") ABOLISHED BY JESUS CHRIST
"You have heard that it was said, 'Eye for eye, and tooth for tooth.' But I tell you, Do not resist an evil person but whoever slaps you on your right cheek, turn to him the other also, and whosoever want to get to sue you and take your tunic, let him have your coat also, and whosoever shall compel thee to go one mile, go with him twain. Give to him that asketh thee, and from him that would borrow of thee turn not thou away" (Matthew 5:...)
The Jews were in the habit of applying the talionis law (law of revenge) in their judgments (eye for eye, tooth for tooth, hand for hand, foot for foot, burn for burn... and so on.), And so they wrote it in the laws of the old Testament (Exodus 21:24-25). The Lord abolished this law of "an eye for an eye and a tooth for a tooth" because it was a law that did not included forgiveness or mercy, then weren't God's Law, because what God likes is the mercy.
"You have heard that it was said, 'Love your neighbor and hate your enemy.' But I tell you: Love your enemies, bless them that curse you, do good to them that hate you, and pray for those who spitefully use you and persecute you.... " (Matthew 5...)
The Jews, often beheaded or hanged his enemies, and not only to them, but then, as says the laws of the Old Testament, entered the conquered peoples and in many cases children and women were slaughtered... Jesus Christ abolished these Old Testament laws and commands us to love our enemies.
Other Old Testament laws abolished by Jesus Christ:
"If a man commits adultery with his neighbor's wife, the adulterer and the adulteress shall surely be dead". (Leviticus 20:10)
"And the daughter of any priest, if she fornicate, dishonor his father; she shall be burnt with fire" (Leviticus 21:9)
That Is not what Jesus did with the adulterous woman in the Gospel Example:
"And the scribes and the Pharisees bring a woman taken in adultery; and having set her in the midst, they say unto him, Teacher, this woman hath been taken in adultery, in the very act. Now in the law Moses commanded us to stone such: what then sayest thou of her? And this they said, trying him, that they might have whereof to accuse him. But Jesus stooped down, and with his finger wrote on the ground.
But when they continued asking him, he lifted up himself, and said unto them, He that is without sin among you, let him first cast a stone at her. And again he stooped down, and with his finger wrote on the ground. And they, when they heard it, went out one by one, beginning from the eldest, even unto the last: and Jesus was left alone, and the woman, where she was, in the midst. And Jesus lifted up himself, and said unto her, Woman, where are they? did no man condemn thee? And she said, No man, Lord. And Jesus said, Neither do I condemn thee: go thy way; from henceforth sin no more. Again therefore Jesus spake unto them, saying, I am the light of the world: he that followeth me shall not walk in the darkness, but shall have the light of life". (Juan 8:3-12)
With this teaching, Jesus abolished the Old Testament commandments that ordered kill people for various reasons. So far we have collected some examples of Old Testament commandments that Jesus abolished, but the argument that Jesus had with the Jews because of the law, is much broader and at this discussion is devoted a large part of the Gospel. Jesus taught us the true law of God.