The house of Israel is the line of Jacob, called Israel, and the people descended from his twelve sons, who became the twelve tribes. In the record the house of Israel is the Lord’s covenant people, scattered across the face of the earth yet armed with righteousness as the saints of the church of the Lamb (1 Nephi 14:14).
The record teaches that after the house of Israel was scattered it would be gathered again, the natural branches grafted back in and brought to a knowledge of the Messiah their Redeemer (1 Nephi 10:14; 19:16). The Book of Mormon’s most extended account of this covenant history is the allegory of the olive tree, drawn from the prophet Zenos and recorded in Jacob 5. In it the house of Israel is the tame olive tree whose natural branches are broken off and planted throughout the vineyard, wild branches are grafted in their place, and in the last days the Lord of the vineyard calls servants to graft the natural branches back so the tree may bear fruit again (Jacob 5:3-14, 71-75). Jacob closes by noting that God remembers Israel “both roots and branches” and stretches forth his hands to them all the day long (Jacob 6:4). Lehi spoke not of his own seed alone but of all the house of Israel, pointing to the covenant the Lord made with Abraham—that in his seed all the kindreds of the earth would be blessed—to be fulfilled in the latter days (1 Nephi 15:18). That covenant is of worth to the Gentiles and to all the house of Israel (1 Nephi 22:9).
Gentiles who repent and come unto Christ are numbered among the house of Israel (2 Nephi 10:18; 3 Nephi 30:2). When they accept the gospel they enter the covenant and are numbered among the remnant of Jacob, who will help build the New Jerusalem (3 Nephi 21:22-23). The Bible, the record of the Jews, contains the covenants the Lord made to the house of Israel (1 Nephi 13:23).
The covenant the Father made with his people is fulfilled when these things come to pass and Jerusalem is again inhabited by his people as their land of inheritance (3 Nephi 20:46).