What did Jesus mean that He came not to abolish the law, but to fulfill it? What did He mean that 'not an iota, not a dot, will pass from the Law until all is accomplished' (Matthew 5:18)?

TL;DR:

Jesus honored the Law as God's Holy Word and declared the Law to be perfect, non-negotiable, and a true prediction of God's work in the future. The fulfillment of the Law is found in Christ Himself and His finished work of redemption.

what does the bible say?

In Luke 16:17 Jesus says, "But it is easier for heaven and earth to pass away than for one dot of the Law to become void." In Matthew 5:17–18, as part of His Sermon on the Mount, Jesus says, "Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them. For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished." It is important to keep Jesus' statements in context. In Luke He said, "The Law and the Prophets were proclaimed until John" (Luke 16:16, NIV). He quickly follows that statement in verse 17 with an affirmation of the Law's permanence, to curtail any thinking that John the Baptist's coming had somehow voided the Law. Even as the kingdom was being preached to the nation of Israel, the Law held its proper place as the revelation of God's holy standard. As John MacArthur puts it, "The great moral principles of the law, the eternal truths contained in the law's types and symbols, and the promises recorded by the prophets all remain in force and are not abrogated by the kingdom message" (Study Helps, The MacArthur Study Bible, 1997).

from the old testament

The Psalms declare that the Law of the Lord is perfect (Psalm 19:7–10; 119:1–8). Many synonyms are used for the Law including testimonies, precepts, statutes, and commandments.
Till heaven and earth pass away, God's Holy Word—all of it—will continue to accomplish all that it was sent to do (Isaiah 55:11). Whether His laws are prophetic or personal, everything God says will come to pass. Those who keep His laws will be blessed.

from the new testament

Keeping the laws of the Old Testament take on new meaning with the advent of Christ. Christians do not work at fulfilling the Mosaic law because Christ fulfilled it on our behalf (Matthew 5:17). The Mosaic covenant was given to the nation of Israel in a specific time and place for a specific purpose, and fulfilled in Christ (see Hebrews). Among other things, the Law points to God’s character and exemplifies the types of ways in which His people are to live. The Law is instructive, useful, and meaningful for us (2 Timothy 3:16–17). But it is not directly applicable in the way it was to ancient Israel.
We are saved by God's grace (Ephesians 2:8–9) not by keeping the Law. The Law was never the way of salvation (Galatians 3:1–29).
Being under grace, believers are not bound by the Law (Romans 6:14). We have the Holy Spirit actually living in our hearts; He empowers us to live for the things of God (Philippians 2:12–13). The Spirit gives us the ability to "love one another" (John 13:34), and we know that "the one who loves another has fulfilled the law" (Romans 13:8). Christians fulfill the "law of Christ" by the Spirit's power (Galatians 6:2).
Jesus introduces the new covenant, ratified with His shed blood (Luke 22:20), but the new covenant does not destroy the Law; it completes the Law.
Jesus gave us a new commandment revealing the Law to be a matter of the heart (John 13:34–35; Matthew 22:37–40).

implications for today

The Law was always meant to reveal our sin and need for the Lord. It gave the Israelites a specific glimpse into who God is and how He wanted them to live. It provided the framework for revealing how we must enter God’s presence. The Law never saved anyone, although it did atone for the people’s sins and pointed to the once-for-all sacrifice that came through Christ. The sacrifices the Hebrew children had to offer every day were not somehow nullified by the death of Christ; they found their fulfillment in His death. We no longer bake bread, sprinkle it with frankincense, and arrange it on a table in a temple (Leviticus 24:5–7). Instead, we look to the Bread of Life, Jesus Christ (John 6:51). The table of showbread was a picture of what was to come; it is no longer part of our worship today, but it remains part of the foundation upon which our faith is built (Ephesians 2:20). Followers of Jesus are saved by grace through faith in His finished work on the cross and His resurrection from the dead (Ephesians 2:1–10). This is the good news, the gospel, that since we cannot keep the Law, but Christ fulfilled it, we have all the blessings of the heavens in Him. With the coming of God’s Spirit, believers have His law written on our hearts (Ephesians 1). Believers seek to live for the Lord and fulfill the law of Christ by walking in God’s gift of forgiveness from sin, abiding in His love, and gradually growing to be more like Him in holiness and love (John 15:1–17).

understand

Jesus fulfilled the prophecies and requirements of the Law, bringing them to their intended end.
The Law’s purpose was to reveal sin and point to the need for redemption, which Jesus accomplished through His sacrifice on the cross.
Jesus introduced the new covenant, fulfilling and transcending the Law.

reflect

How does understanding that Jesus fulfilled the Law rather than abolished it impact your view of its relevance in your life today?
How can you reflect the heart-centered obedience Jesus taught, as opposed to just following rules for their own sake?
How does knowing that Christ's fulfillment of the Law means you are under grace influence your approach to personal righteousness and spiritual growth?

engage

Some people struggle to understand the Old Testament way of salvation. However, salvation was always by God’s grace through faith. Hebrews 11 reveals several people in the Old Testament whose faith was counted for righteousness, revealing that salvation always came by faith in God.
The Law was given to the Israelites to reveal who God is and how He was calling them to live. The Law was fulfilled by Christ, but the Law did not disappear. We still follow aspects of the Law that are repeated in the New Testament, reaffirmed by Jesus, or that are part of overall moral principles or the heart of the Law. The Law is also helpful to provide contextual insights into God’s character and the general ways in which He calls His people to live. Though the Mosaic law is not directly applied to us today, it still has much relevance in our lives.
How can recognizing that the Law's requirements were met in Christ shape your understanding of the balance between grace and obedience in the Christian walk?