I am glad you asked me that, because some people think that when I say "old covenant" I refer to the promises that God made to "Abraham and his seed", as these are promises that still continue in the new covenant through "his seed" which is Christ (
Galatians 3:16) in whom all the families of the earth are blessed.
The old covenant that the children of Israel were under is the one made at Mt. Sinai.
This is the covenant where the law of Moses was given, and was broken by the children of Israel.
This is also the covenant that bound the children of Israel to the holy land.
When God describes the blessings for obedience and the curses for disobedience to Israel, one of the cursed includes
Deuteronomy 28:21 "The Lord shall make the pestilence cleave unto thee, until he have consumed thee from off the land, whither thou goest to possess it"
A good illustration of the old and new covenants:
Galatians 4:22 For it is written, that Abraham had two sons, the one by a bondmaid, the other by a freewoman.
23 But he who was of the bondwoman was born after the flesh; but he of the freewoman was by promise.
24 Which things are an allegory: for these are the two covenants; the one from the mount Sinai, which gendereth to bondage, which is Agar.
25 For this Agar is mount Sinai in Arabia, and answereth to Jerusalem which now is, and is in bondage with her children.
26 But Jerusalem which is above is free, which is the mother of us all.
27 For it is written, Rejoice, thou barren that bearest not; break forth and cry, thou that travailest not: for the desolate hath many more children than she which hath an husband.
28 Now we, brethren, as Isaac was, are the children of promise.
29 But as then he that was born after the flesh persecuted him that was born after the Spirit, even so it is now.30 Nevertheless what saith the scripture? Cast out the bondwoman and her son: for the son of the bondwoman shall not be heir with the son of the freewoman.
31 So then, brethren, we are not children of the bondwoman, but of the free.
Notice that despite having broken the old covenant, the children of Israel have a far superior hope in the new covenant.
It is an eternal spiritual hope shared by all in Christ, but it is not found in the earthly kingdom.