Chapter 16 vs 18 Appoint judges and officials for yourselves from each of your tribes in all the towns the  Lord your God is giving you. They must judge the people fairly.  19-20 You must never twist justice or show partiality. Never accept a bribe, for bribes blind the eyes of the wise and corrupt the decisions of the godly. Let true justice prevail, so you may live and occupy the land that the Lord your God is giving you.

Chapter 17 vs 8-9 Suppose a case arises in a local court that is too hard for you to decide - for instance, whether someone is guilty of murder or only of manslaughter, or a difficult lawsuit, or a case involving different kinds of assault. Take such legal cases to the place the Lord your God will choose, and present them to the Levitical priests or the judge on duty at that time. They will hear the case and declare the verdict. 10-13 You must carry out the verdict they announce and the sentence they prescribe at the place the Lord  chooses. You must do exactly what they say.  After they have interpreted the law and declared their verdict, the sentence they impose must be fully executed; do not modify it in any way. Anyone arrogant enough to reject the verdict of the judge or of the priest who represents the Lord your God must die. In this way you will purge the evil from Israel.  Then everyone else will hear about it and be afraid to act so arrogantly. Deuteronomy 16:18-20 & 17:8-13 New Living Translation paraphrase. (English Standard Version link)

If these were the people God had saved out of Egypt and to whom He had given such good laws to live by, why would they need judges, and why do you think Moses said the words we read in vs 19-20?

The answer must be because they still had hearts that were so ready to do wrong. Many years later God spoke through the prophet Ezekiel and said: “I will give you a new heart, and a new spirit I will put within you. And I will remove the heart of stone from your flesh and give you a heart of flesh. And I will put my Spirit within you, and cause you to walk in my statutes and be careful to obey my rules.” (Ezekiel 36:26-27) That promise was ultimately fulfilled when Jesus sent the Holy Spirit to His Church, and why Christians are called to live as the Holy Spirit leads and not according to wrong desires that still arise in our thoughts.

Sadly, quarrels and disputes can still arise even among Christians. In Corinthians 6:1 & 3 Paul writes: “When one of you has a dispute with another believer, how dare you file a lawsuit and ask a secular court to decide the matter instead of taking it to other believers! Don’t you realize that we will judge angels? So you should surely be able to resolve ordinary disputes in this life.” Being saved from the condemnation our sins deserve doesn't mean we are suddenly perfect in our behaviour. But the Holy Spirit has come into our life to guide us in God's will and help us to put His commandments first in our lives.

What was the important matter that vs 10-13 dealt with?

Having pointed out that some issues might be too difficult for a local judge and should be taken to the Levitical priests at a place that God would appoint (like the Supreme Court in our days) Moses warns that whatever judgement was made needed to be obeyed to the letter. The Levitical priests in those days were speaking for God and so their verdict was to be followed. The people were not allowed to organize protests and marches if they disagreed with the judgement. This shows how vitally important vs 19-20 were, even for the Levitical priests! These things are an Old Testament example of what the apostle's said to Christians concerning those whom God has placed in authority - even ungodly ones.

Romans 13:1-2 says: “Submit to governing authorities, for all authority comes from God, and those in positions of authority have been placed there by God. So anyone who rebels against authority is rebelling against what God has instituted, and they will be punished.” And 1 Peter 2:13 says: “For the Lord’s sake, submit to all human authority - whether the king as head of state, or the officials he has appointed. For the king has sent them to punish those who do wrong and to honour those who do right.” Obviously we’re not to do evil if the the authorities demand that – we need then to obey God first – but in general we are called to obey those God has allowed to be in power, and especially to respect law, order and justice.