How Does Jesus Fulfill the Law?

One of the greatest challenges in studying and teaching the Bible is understanding how the new covenant relates to the old. Jesus said he didn’t come to abolish the law or the prophets, but to fulfill them (Matt. 5:17). What did he mean by this? Does he fulfill all Old Testament laws in the same way? What difference does Jesus’s fulfillment of the law make to how we read and apply the Old Testament as Christians?