The Abraham Covenant was initiated by God and the parties involved was God and Abram the Hebrew (Gen. 14:13) and his Hebrew seed/descendants (this covenant is everlasting):
7 And I will establish my covenant between me and thee and thy seed after thee in their generations for an everlasting covenant, to be a God unto thee, and to thy seed after thee. Genesis 17:7.
The Mosaic Covenant was initiated by God and the parties involved was God and the children of Jacob/Israel. This covenant was later superseded by the New Covenant prophesied by the prophet Jeremiah in 31:31-34. This succession did not make the Mosaic Covenant 'obsolete' or 'abolished' but was fulfilled by Christ and through our being 'born-again' the Mosaic Covenant between God and the children of Israel took on a new dynamic. Of the three parts of the Mosaic Covenant were the Moral Laws (Ten Commandments), the Social Laws, and the Ceremonial Laws, of which the Ceremonial Laws have been fulfilled by the sacrificial offering for sin of Jesus Christ, the lamb of God.
5 Now therefore, if ye will obey my voice indeed, and keep my covenant, then ye shall be a peculiar treasure unto me above all people: for all the earth is mine:
6 And ye shall be unto me a kingdom of priests, and an holy nation. These are the words which thou shalt speak unto the children of Israel. Exodus 19:5–6.
The New Covenant was initiated by God and the parties involved are God and the House of Israel (ten northern kingdom tribes), and the House of Judah (two southern kingdom tribes.) This covenant was initiated by Christ at His last Passover and came into effect with the advent of the Holy Spirit of Promise on the Jewish Feast of Harvest (Pentecost ca. AD 32.)
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;
Which my covenant they brake,
Although I was an husband unto them, saith the LORD:
33 But this shall be the covenant that I will make with the house of Israel;
After those days, saith the LORD,
fI 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.
34 And they shall teach no more every man his neighbour (member of a neighboring tribe), and every man his brother (member of the same tribe), 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.
Jeremiah 31:31–34.
