Obligation to God
"You must love the Lord your God with all your heart, with all your soul, and with all your mind"(Matthew 22:37).
Obligation to Man
"The second is like it: "You must love your neighbor as yourself" (Matthew 22:39).
The first four commandments, on the first tablet, outline man's obligations to God. The second six commandments, on the second tablet, reveal man's obligations to his fellow man. Biblically, the number six is frequently used to refer to a man.
The Decalogue
The ten commandments are referred to as the decalogue because there are, famously, ten laws. God wrote the decalogue on tablets of stone using His own finger. This was done to leave no doubt as to the author.
The tablets were given to Moses on Mount Sinai. The type of stone chosen was quite likely lapis lazuli. This stone was very valuable in ancient times. Lapis lazuli is a beautiful sky blue stone which has the appearance of being sprinkled with gold. Blue is often associated with the word of God and symbolizes truth—hence the phrase, "true blue." Many Bible translations prefer sapphire. God had an ample supply of this stone on the mountain. The pavement under His throne is as clear blue as the sky on a brilliant sunny day.

And they saw the God of Israel: and there was under his feet as it were a paved work of a sapphire stone, and as it were the body of heaven in his clearness. . Exodus 24:10
The Lord spoke the ten commandments to the people from Mount Sinai with a terrifying display of authority. His words were accompanied by lightnings, thunder, fire and earthquakes. A long trumpet blast waxed louder and louder to an almost deafening pitch. That day, on the 6th of Sivan, the people made a covenant with God to obey His commandments. This agreement is referred to biblically as the first covenant or "old covenant."

And he took the book of the covenant, and read in the audience of the people: and they said, All that the LORD hath said will we do, and be obedient. Exodus 24:7
After this, Moses climbed the mountain to receive the tablets. He was with the Lord for 40 days and 40 nights fasting. When he returned, the people had made a golden calf to worship. So quickly, every precept of the first covenant had been broken and there was need for a new covenant. Moses broke the first set of tablets with righteous indignation.
Once again, Moses climbed the mountain for the second set of tablets and remained another 40 days. Unlike the ceremonial law which was written on skins and placed beside the ark of the covenant, the tablets of stone were placed under the mercy seat inside the ark. The ark was then placed in the most holy compartment of the sanctuary.
Can we Keep the Ten Commandments?
Some teach the 10 commandments were nailed to the cross with Christ and no longer apply. Actually, only ceremonial sacrifices, circumcision and the record of our pardoned transgressions are nailed to that tree. The ten commandments are eternal laws that have always existed and will continue to be the standard forever.
If You Love Me, Keep My Commandments
As a result of sin, we are not naturally inclined to keep God's commandments. Even converted Christians cannot obey God's law right away. Only love for God fosters obedience. This kind of love is progressive and is the result of contemplating God's monumental sacrifice. A deeper understanding of Christ's grace draws us to Him and inspires us to obedience.
And all the people saw the thunderings, and the lightnings, and the noise of the trumpet, and the mountain smoking: and when the people saw it, they removed, and stood afar off.
And I, if I be lifted up from the earth, will draw all men unto me. John 12:32
The New Covenant
As we draw closer to Jesus, our heart of stone is gradually replaced with a heart of flesh capable of responding to love. Only God can change the heart and desires. The law of God convicts us of sin but is powerless of itself to effect change. We cannot be obedient of our own through works. Obedience to God is the fruit of grace. We are being sanctified step by step through a change of mind and heart.

For by one offering he hath perfected for ever them that are sanctified. Whereof the Holy Ghost also is a witness to us: for after that he had said before, This is the covenant that I will make with them after those days, saith the Lord, I will put my laws into their hearts, and in their minds will I write them. Hebrews 10:14-16
The first covenant could not succeed because obedience was through drudgery of works. The new covenant succeeds because obedience becomes desirable. God's laws are written on the heart and mind. Since Adam, the way of obedience has always been through the second covenant. Obedience to God's moral laws has always been the fruit of grace.

Thou shalt love thy neighbour as thyself. Love worketh no ill to his neighbour: therefore love is the fulfilling of the law." Romans 13:9-10