Who is the remnant of Israel?
The remnant of Israel is the portion of the Jewish nation that survives the wrath of the antichrist and the judgments of the tribulation, and who will be converted to faith in Jesus as Messiah just before the Millennium begins.
Scripture speaks of a remnant of Israel that will be preserved through the terrible judgments that come upon the earth. Isaiah 11:11 tells how the Lord will set His hand again to recover the remnant of His people from the nations where they have been scattered, and this is the group with whom His promise of salvation will be fulfilled. They are not the whole nation as it exists today, but those who live through the wrath of the antichrist and the judgments described in Isaiah 24, where the earth is emptied and only a few are left alive.
This remnant is described as being converted in a single day. Zechariah 3 speaks of the Lord removing the iniquity of the land in one day, and Micah 7 speaks of God turning again in compassion, subduing iniquities and casting sins into the depths of the sea. This conversion is understood to happen just before the Millennium starts, since it is this remnant who will serve as keepers of the Temple and offer sacrifices before the Lord, something they could not do unless purged of their sins.
The remnant of Israel is distinguished clearly from the Church. They are the natural seed of Abraham, the ones blessed by God through the covenant, not people who are born again as the sons of God are. During the Millennium they will build houses and inhabit them, plant vineyards and eat the fruit of them, and their offspring will continue after them, enjoying the long, fruitful life of that age. Ezekiel 36 describes how the Lord will cleanse them of their iniquities and cause the desolate land to be rebuilt and inhabited again, becoming like the garden of Eden.
So the remnant of Israel is the surviving, believing portion of the Jewish nation who come through the tribulation, turn to the Lord in the day appointed, and go on to dwell in the land during the Millennium as the earthly people of God, distinct from the glorified sons of God who reign with Christ.
The scriptures
And it shall come to pass in that day, that the Lord shall set his hand again the second time to recover the remnant of his people, which shall be left, from Assyria, and from Egypt, and from Pathros, and from Cush, and from Elam, and from Shinar, and from Hamath, and from the islands of the sea.
Hear now, O Joshua the high priest, thou, and thy fellows that sit before thee: for they are men wondered at: for, behold, I will bring forth my servant the BRANCH. For behold the stone that I have laid before Joshua; upon one stone shall be seven eyes: behold, I will engrave the graving thereof, saith the LORD of hosts, and I will remove the iniquity of that land in one day. In that day, saith the LORD of hosts, shall ye call every man his neighbour under the vine and under the fig tree.
Who is a God like unto thee, that pardoneth iniquity, and passeth by the transgression of the remnant of his heritage? he retaineth not his anger for ever, because he delighteth in mercy. He will turn again, he will have compassion upon us; he will subdue our iniquities; and thou wilt cast all their sins into the depths of the sea.
The earth mourneth and fadeth away, the world languisheth and fadeth away, the haughty people of the earth do languish. The earth also is defiled under the inhabitants thereof; because they have transgressed the laws, changed the ordinance, broken the everlasting covenant. Therefore hath the curse devoured the earth, and they that dwell therein are desolate: therefore the inhabitants of the earth are burned, and few men left.
And they shall build houses, and inhabit them; and they shall plant vineyards, and eat the fruit of them. They shall not build, and another inhabit; they shall not plant, and another eat: for as the days of a tree are the days of my people, and mine elect shall long enjoy the work of their hands. They shall not labour in vain, nor bring forth for trouble; for they are the seed of the blessed of the LORD, and their offspring with them.
Thus saith the Lord GOD; In the day that I shall have cleansed you from all your iniquities I will also cause you to dwell in the cities, and the wastes shall be builded. And the desolate land shall be tilled, whereas it lay desolate in the sight of all that passed by. And they shall say, This land that was desolate is become like the garden of Eden; and the waste and desolate and ruined cities are become fenced, and are inhabited. Then the heathen that are left round about you shall know that I the LORD build the ruined places, and plant that that was desolate: I the LORD have spoken it, and I will do it.
For in mine holy mountain, in the mountain of the height of Israel, saith the Lord GOD, there shall all the house of Israel, all of them in the land, serve me: there will I accept them, and there will I require your offerings, and the firstfruits of your oblations, with all your holy things.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.