About 600 years before Jesus (יֵשׁוּעַ‎), the Hebrew prophet Jeremiah (יִרְמְיָ֖הוּ) said there would be a New Covenant. The Jewish Publication Society (1917) translated him this way:

31:31 "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;

32 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; forasmuch as they broke My covenant, although I was a lord over them, saith the Lord."
(Some translations say, "even though I was married to them"!).

We can see in verse 32 that the reason the Lord was going to make a new covenant was because Israel and Judah (the north and south parts of the nation) broke the old one. This agrees with what Moses himself had said would happen, in Deuteronomy (Devarim דְּבָרִים‎ ) 29 and 30:

29:25 "Then the nations will say, 'It is because they have forsaken the covenant of the Lord God of their fathers, which He made with them when He brought them out of the land of Egypt,'"

30:1 "Now it shall come to pass when all these things come upon you, the blessing and the curse which I have set before you, and you call them to mind among all the nations where the Lord your God drives you,

2 and you return to the Lord your God... with all your heart and with all your soul...

6 the Lord your God will circumcise your heart and the heart of your descendants, to love the Lord your God with all your heart and with all your soul..."

So it seems the Children of Israel broke the earlier covenant because our hearts were not right. This is the basic problem of all humanity, not Israel only.

Going back to Jeremiah chapter 31, if our hearts are the problem, how will this new covenant make things any better? Here are what verses 33 and 34 say,

31:33 "But this is the covenant that I will make with the House of Israel after those days, says the Lord: I will put My instruction in their minds, and write it on their hearts; and I will be their God, and they shall be My people.

34 No more shall every man teach his neighbor, and every man his brother, saying, 'Know the Lord,' for they shall all know Me, from the least of them to the greatest of them, says the Lord. For I will forgive their iniquity, and their sin I will remember no more."

This New Covenant consists of letting the Lord write His ways on our hearts, so they become a delight and our desire!  He changes us, from the inside out.

As a loving father, He did this by sending Himself into our time and 3-dimensional space, in the form of the man Yeshua. When we are given the gift of faith (emunah אֱמוּנָה), His Holy Spirit (Ruakh haKodesh  רוּחַ הַקֹּדֶשׁ) comes to live in us to be our help and to guide us into all truth.