Jesus taught us to fulfil the law by love

MATTHEW 22:34-40
34 But when the Pharisees had heard that he had silenced the Sadducees, they were all gathered together.
35 Then one of them, who was a lawyer, asked him a question, tempting him, and saying,
36 Master, which is the great commandment in the law?
37 Jesus said to him, You shall love the Lord your God with all your heart, and with all your soul, and with all your mind.
38 This is the first and great commandment.
39 And the second is like it, You shall love your neighbour as yourself.
40 On these two commandments hang all the law and the prophets.

ROMANS 13:8-10 (Paul)
8 Owe no man anything, but to love one another: for he who loves another has fulfilled the law.
9 For this, You shall not commit adultery, You shall not kill, You shall not steal, You shall not bear false witness, You shall not covet; and if there is any other commandment, it is briefly comprehended in this saying, namely, You shall love your neighbour as yourself.
10 Love works no ill to his neighbour: therefore love is the fulfilling of the law.

GALATIANS 5:14 (Paul)
14 For all the law is fulfilled in one word, even in this; You shall love your neighbour as yourself.

Note: Does fulfilling of the law mean that we do not have to keep the ten commandments? No, because Jesus said, "all the law" (Matthew 22:40) hangs on the commands to love, which includes all of the ten commandments, not just nine of them. Secondly, we fulfil the law by love (Matthew 22:34-40; Romans 13:8-10; Galatians 5:14), and if we love God with all of our heart and soul, and our neighbour as ourselves, then our love for God is expressed by obedience to his commandments:

(John 14:21-24) "He who has my commandments, and keeps them, he it is who loves me: ... If a man loves me, he will keep my words: ... He who does not love me does not keep my sayings: and the word which you hear is not mine, but the Father's which sent me."
(1 John 5:2-3) "By this we know that we love the children of God, when we love God, and keep his commandments. For this is the love of God, that we keep his commandments: and his commandments are not grievous."
(2 John 6) "And this is love, that we walk after his commandments."

This is why Jesus taught us two commandments which enable us to fulfil the law:

(Matthew 22:37) "You shall love the Lord your God with all your heart, and with all your soul, and with all your mind."
(Matthew 22:39) "You shall love your neighbour as yourself."

The latter one of these two gives us the desire to keep the last six of the ten commandments, whereas faith would give us the ability to keep them. For example:
If we love our father and our mother, would we dishonour them? No, because we would hate to do such a thing, and we would only want to please them.
If we love our neighbours would we kill them? or anyone that they love? No, because we would hate to do such a thing, and we would only want to do what is best for them. We would not hate them if they offended us, because hate is spiritual murderer (1 John 3:15), and love is the opposite of hate. We would not speak evil of them, because spiritual death is in the power of the tongue (Proverbs 18:21), and we would not wish to hurt them.
If we love our neighbours, would we steal from them? No, because we would hate to do such a thing, and we would only want to do what is best for them. Love is a desire to give (John 3:16), and would give us the opposite desire to stealing from them. We would never wish to see them suffer the hurt from our actions, so love would motivate us to keep this commandment.
If we love our neighbours would we lie to them? No, because the scripture declares, "A lying tongue hates those who are afflicted by it;" (Proverbs 26:28). Paul declared, "Therefore putting away lying, speak every man truth with his neighbour:" (Ephesians 4:25), because speaking the truth to someone is an expression of love. So fulfilling this commandment by love would mean that we would never lie to our neighbour.
So likewise, love for our neighbours would keep us from breaking any of the commandments that apply to them.
In the same way, if we love God with all of our heart, and with all of our soul, and with all of our mind, would we put other Gods before him? If you have a wife or a girl friend that you love with your whole heart, would you put other women or girls before them? No way! Love would keep you from it.
Would we make carved images that are an abomination to God, and bow down to them, knowing how much this would hurt God? No way! because our love for God would keep us from it.
Would we take his name in vain, knowing that it would hurt him? No way!
How then would we not keep the Sabbath, which he made for us (Mark 2:27), knowing that this would be like throwing his gift back in his face?
So there it is, love fulfils the law, and love expresses itself as obedience to God's commandments. Jesus said, "He who does not love me does not keep my sayings;" (John 14:24), and one of his sayings was, "keep the commandments." (Matthew 19:17), so people who do not keep the ten commandments do not fully love Jesus or God.

Jesus came to fulfil the law

                MATTHEW 5:17-18 (Jesus)

17 Do not think that I am come to destroy the law, or the prophets: I am not come to destroy, but to fulfil.
18 For amen I say to you, Till heaven and earth pass, one jot or one tittle shall in no way pass from the law, till all is fulfilled.

Note 1: Notice that Jesus said, that he did not "come to destroy the law ... but to fulfil." (Matthew 5:17), and from the words of Jesus himself we can know that he did fulfil as much as he could at his first coming, because he said, "I have kept my Father's commandments." (John 15:10). Also when praying to his Father before his death, he said, "I have finished the work which you gave me to do." (John 17:4), and just before he died, he said, "It is finished:" (John 19:30). There are some things, such as the feast of trumpets, for example, which Jesus will fulfil at his second coming (Matthew 24:31; 1 Corinthians 15:52; 1 Thessalonians 4:16), but this does not mean that he has not fulfilled as much of the law as was expected of him. Does the fact that Jesus fulfilled the law mean that we do not now have to fulfil it? No, that cannot be so because we are expected to be like Jesus:

(Matthew 10:25) "It is enough for the disciple that he be as his master, and the servant as his Lord."
(Romans 8:29) "For whom he did foreknow, he also did predestine to be conformed to the image of his Son,"
(1 Corinthians 15:48) "as is the heavenly, such are they also that are heavenly."
(1 Peter 2:21) "For even to this you were called: because Christ also suffered for us, leaving us an example, that you should follow his steps:"
(1 John 2:6) "He who says he abides in him aught himself also to walk, even as he walked."
(1 John 4:17) "In this is our love made perfect ... as he is, so are we in this world."

If Jesus fulfilled the law, and we are meant to "be as our master", "be as our Lord", "be conformed to his image", "follow his steps", "walk, even as he walked", and be "as he is", then we are meant to fulfil the law also as Jesus did. Again there are exceptions, such as the sin offering (Leviticus 16:15-16), and the scapegoat on the day of atonement (Leviticus 16:20-23), where Jesus carried away the sins of the world. That cannot be done by us, but we do need to fulfil that which we can fulfil:

(Romans 8:4) "That the righteousness of the law might be fulfilled in us, who walk not according to the flesh, but according to the spirit."

Note 2: Notice that Jesus also said, "Till heaven and earth pass, one jot or one tittle shall in no way pass from the law, till all is fulfilled." (Matthew 5:18). There are two separate conditions here, and nothing of the law will pass away until at least one of these is fulfilled. They are, "Till heaven and earth pass", or "till all is fulfilled". Has heaven and earth passed yet? No:

(2 Peter 3:7-10) "But the heaven and earth which are now, by the same word are kept in store, reserved for fire against the day of judgement and perdition of ungodly men. ... the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are in it shall be burned up."

So this condition has not removed anything from the law yet.

Note 3: The other condition, "till all is fulfilled", can be taken in two different ways, one of which is, it can mean "till all (the law) is fulfilled" (See #1.25). However, if we examine the Greek, ewj an panta genhtai, (Gtr. heos an panta genetai), which also occurs elsewhere in scripture (Luke 21:32), and where it is also translated "till all is fulfilled", we can compare it with other parallel scriptures:

(Matthew 24:34) "till all these things are fulfilled."
(Mark 13:30) "till all these things are done."

Taking these latter three scriptures in context, it seems obvious that they all refer to the events of the end time, before Jesus returns, things which have not happened to the Jews yet. So some could conclude that Jesus' statement "till all is fulfilled" (Matthew 5:18), literally could mean that "one jot or one tittle shall in no way pass from the law, until all (end-time events) are fulfilled", in which case, the law must still be completely intact. This being so, the Sabbath day commandment must also be intact, and we still need to fulfil it.