"Do not think that I came to abolish the Law or the Prophets; I did not come to abolish but to fulfill. For truly I say to you, until heaven and earth pass away, not the smallest letter or stroke shall pass from the Law until all is accomplished. Whoever then annuls one of the least of these commandments, and teaches others to do the same, shall be called least in the kingdom of heaven; but whoever keeps and teaches them, he shall be called great in the kingdom of heaven. For I say to you that unless your righteousness surpasses that of the scribes and Pharisees, you will not enter the kingdom of heaven" (Matthew 5:17-20).
What does this passage mean? I don't think it's all that obvious.
In 5:17, most Christians would read this as Jesus pointing out that he has no intention of doing away with the Law. However, in the context of the time, he is saying that he didn't come to misrepresent or distort the Torah. Rather, he came to give a new - and correct - understanding of Torah, to show the true meaning. That's understanding this verse as the terms were used in discourse in First Century Israel. (It's important to remember that "law" is Torah, a unit encompassing much more than individual rules: It's more properly understood as God's teaching, not God's law.)
Then, the key phrase in understanding the role of the Law today is "until all is accomplished" (v.18). To understand this, we need to know what Jesus set out to do. What did he come to accomplish? It was to open the way for the redemption of Creation and the reconciliation of humankind - you and me - with our God. So, when that was done, when the door was open for all to come in, then, regarding Jesus' life on earth, "all was accomplished." So when was that?
Certainly, it was not at his birth. He was speaking long after his birth, right? But he was speaking before the cross. So was all accomplished at the cross? No, because the cross needed the resurrection to attest to its authenticity. So, had Jesus ascended after the resurrection, and God had done nothing more to bring reconciliation, could the world be reconciled? Probably not. It would have been technically possible, but nobody would have known about it.
After the resurrection, Jesus ascended, and his in-person work here was ended. But clearly there was much remaining to be done. Nobody outside a small, insignificant area, knew. So what was lacking? Pentecost. At that time, the Spirit indwelled all believers - the first time ever - and empowered them to carry on with the implementation of what Jesus began. At that point, all that Jesus began was accomplished. At that time, the prophecy in Jeremiah 31:33 - a new heart - began its fulfillment, and Ezekiel 11:19 and 36:26 also had their beginnings.
So now, the Law is done. It did its job. All that Jesus came to do has been completed, and the Law is no longer necessary. Further, those who would make it of value for today miss that the Law was about obeying rules out of a love for God. Today, he calls us to know his heart. He calls us his friends. He fills us with the Holy Spirit so that his law is, indeed, written on our hearts, as we listen to the whisperings of his Spirit.