Hebrews 8:10-11

10 For this is the covenant that I will make with the house of Israel after those days, saith the Lord; I will put my laws into their mind, and write them in their hearts: and I will be to them a God, and they shall be to me a people:

11 And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest.
KJV

This is a reference back to prophecy in Jeremiah 31:31-34

"Behold, the days come, saith the LORD, that I will make a new covenant with the house of Israel, and with the house of Judah:  Not according to the covenant that I made with their fathers in the day that I took them by the hand to bring them out of the land of Egypt; which my covenant they brake, although I was an husband unto them, saith the LORD:  But this shall be the covenant that I will make with the house of Israel; After those days, saith the LORD, I will put my law in their inward parts, and write it in their hearts; and will be their God, and they shall be my people.  And they shall teach no more every man his neighbour, and every man his brother, saying, Know the LORD: for they shall all know me, from the least of them unto the greatest of them, saith the LORD; for I will forgive their iniquity, and I will remember their sin no more.

God is contrasting the two covenants.  To understand this scripture we need to look at the differences between the two. 

It is important to note here that the Israelites were physically born into the old covenant while Christians today are spiritually born into the new covenant.  Because of this all manner of people whether faithful or not were legally associated with the old covenant simply because they were Israelites.  There is nobody like that under the new covenant.  Only those who know Jesus, know the word and have obeyed the gospel are called God's children today.  See John 1:12. 

Everybody under the new covenant who are called the children of God know the Lord, from the least to the greatest.  Knowing Him is essential for becoming a Christian.  There are no Christians who do not know God. 

Putting His laws into their minds and writing them on their hearts is a way of stressing the spiritual nature of the new covenant.  The old law was full of numerous ordinances of service which were tedious and had to be observed with the strictest of attention to detail.  The purpose of this was to demonstrate just how serious sin was by the impossibility of working enough to be forgiven.  God literally made them jump through hoops and this was to teach them and us just how distasteful sin is to God.  And this was made evident and driven home when God demanded the death of His Son for the forgiveness of sin. 

God does not write His laws in our minds today or in our hearts by some kind of supernatural means as some would suggest.  Looking at some of the characteristics of the word, we can discern what is meant here by the inspired writer.  "In the beginning was the word and the word was God" (John 1:1).  "And the word was made flesh" (John 1:14).  "The word of God is living, and active" (Hebrews 4:12).  "It is the spirit that quickeneth; the flesh profiteth nothing: the words that I speak unto you, they are spirit, and they are life" (John 6:63).  The ink and the paper and binding of the Bible is not what is in view here.  When God's laws are written  in our minds and on our hearts through His living word, then it is written there by God Himself.