Concise Bible Dictionary:

This word constantly occurs in the Old Testament in the sense of “the rest of the people.” In every crisis in the history of Israel there has been a remnant; this was seen in the time of Ahab (1 Kings 19:18), and so too in the introduction of Christianity (Luke 2:38), and that it will be so in the future is abundantly evident from the testimony of the prophets. There will be great prosperity in the land, and God will cause the remnant of His people to possess it (Zech. 8:12; compare Rev. 12:17). When God’s people are unfaithful to His calling, He secures His own purpose in a remnant.
The prophetic language in the Psalms is not that of the mass of Israel, but of the remnant, in whom the Spirit of Christ speaks; and it is in the Psalms that the remnant is first seen as distinguished from the ungodly nation. The idea of a remnant is found also in the address to the church in Thyatira, and to that remnant (“the rest”) it was said, “That which ye have already hold fast till I come” (Rev. 2:24-25). They represent the faithful in the time of the supremacy of the apostate Popish system.
A remnant represents morally the original whole, and does not imply an inferior remaining portion. It is of God’s grace that any are enabled to be steadfast to the original truth and calling during a general apostasy from it.

From Anstey’s Doctrinal Definitions:

This term means “a small portion of the people” or “the rest of the people.” It is used in a few different ways in Scripture. It may be in connection with Israel’s history (2 Kings 19:4, 30-31; Isa. 1:9; Ezra 9:8; Neh. 1:3), or in connection with the Jews (Rev. 12:17) and the ten tribes of Israel in a coming day (Isa. 10:20-22; 11:11, 16). Or, it could be used in reference to believers in the Christian testimony today (Rev. 2:24 – “the rest [remnant]” W. Kelly Trans.).
As to Israel in a coming day, God has promised that He will not cast off the nation forever. The governmental blindness that is over their hearts will be lifted, and God’s blessing will be upon them when the Lord appears “with healing in His wings” (Mal. 4:2; Rom. 11:25; 2 Cor. 3:14-16). But Scripture does not teach that this applies to every single Israelite by birth. As mentioned earlier, the redemption of Israel will be accomplished in a remnant of the people only. Romans 11:26 states that “all Israel shall be saved” and blessed of God, but previously in chapter 9:6-8, Paul explained that “not all are Israel which are of Israel.” Thus, to be a true Israelite, one needs not only to be of Abraham’s bloodline, but also to have Abraham’s faith. Many of Abraham’s descendants are the “seed” of Abraham, but they are not the “children” of Abraham, and thus, not “the children of God.” (Compare Romans 2:28-29.) Hence, when Paul said that “all” Israel would be saved, he was assuming that the reader has been following the logic of his discourse in the epistle concerning this point. The “all” in Romans 11:26, therefore, is all who are true Israelites—inwardly and outwardly. This shows the importance of reading Scripture is context.
The prophetic Scriptures indicate that there will be two parts to the remnant, or perhaps two remnants: a remnant of Jews (the two tribes) and a remnant of the ten tribes of Israel. These will be brought together in one when the Lord appears and restores the nation, and they will inherit the kingdom together (Isa. 11:12-13; Ezek. 37:15-19, etc.). To have an understanding of God’s dealings with Israel, the student of prophecy needs to distinguish these two parts of the nation. Three main differences are:
The Jews (the two tribes) will return to their homeland in the early part of Daniel’s 70th week (Dan. 9:27) before the Great Tribulation (Isa. 18:1-4), and a remnant will distinguish themselves among the mass as having faith and the fear of God (Isa. 8:16; 66:2; Psa. 1:1-3). The ten tribes will not come back to the land until after the Great Tribulation is over and the King of the North has devastated the land (Matt. 24:29-31). While the mass of the ten tribes will return to their homeland, only a remnant of them will be found to have real faith and will be allowed to enter the land (Ezek. 11:9-11; 20:34-38; Amos 9:9-10).
The Jews have rejected Christ (John 19:15) and will receive the Antichrist (John 5:43), but the ten tribes are guilty of neither. The ten tribes were not in the land when Christ came unto His own and His own received Him not (John 1:11). Nor will they be in the land when the Antichrist arises and is received by the Jews.
The remnant of the Jews will confess “bloodguiltiness” in the crucifixion of Christ and will also own the breaking of the Law (Psa. 51:14; Acts 7:53), but the remnant of the ten tribes will confess their failure in departing from the Law and turning to idolatry (Isa. 26:13; Hos. 14:8).
There are going to be a number of wise and instructed ones raised up among the God-fearing Jewish remnant who will be enlightened in the ways of God and His dealings with the nation. They are referred to as “the Maschilim” (Dan. 12:3; compare also Dan. 11:33). These intelligent ones will act as instructors to the others and will “turn many to righteousness.” Their understanding will probably be such that they will see the Lord Jesus Christ, whom the nation crucified, as the true Messiah of Israel, and will encourage the others to receive Him as such. The remnant of the Jews will eventually do this when Christ appears and they “look on Him whom they pierced” (Zech. 12:10; John 19:37). At that time, they will mourn in repentance (Zech. 12:11-14), and the Lord will open a “fountain” for their cleansing whereupon they will be restored to Him (Zech. 13:1).
There are two parts to the faithful Jewish remnant: a preserved portion that will be miraculously spared from the attacks of their persecutors and from other dangers and will enter the kingdom of Christ on earth—the Millennium (Rev. 12:6, 13-17; 14:1; Psa. 91). Then there will be a martyred portion of the remnant which God will allow to be killed for their faithful testimony during the seven years of tribulation (Rev. 6:9-11; 11:2-12; 14:23a; 15:2-4; Isa. 57:1-2). These godly Jews will be raised at the end of the Great Tribulation and will have a better portion in heaven—being glorified (Rev. 14:13). They will reign with Christ over the earth with the Old Testament saints and the Church in the Millennium (Rev. 20:4; Dan. 7:18, 22, 27).
Since there are these differences of responsibility between the Jews and the ten tribes, the Lord will deal with them differently in view of their restoration. The result will be the same in both—true repentance and a full restoration to the Lord, whereupon His kingdom blessings will be showered upon them. They will constitute the nation of Israel in the Millennium.
The mass of the unbelieving Jews who are not true children of Abraham will be destroyed by the King of the North and his Arab confederacy (Psa. 83:1-8; Psa. 79:1-3; Dan. 2:40-42; Zech. 13:8-9). The mass of the ten tribes of Israelites who are not true children of Abraham will return to the land under the aid of the angels of God (Matt. 24:31) and through the help of some Gentile nations who will fear God (Isa. 14:1-2; 49:22; 60:9; 66:19-20). They will be brought to “the border” of the land of Israel where the Lord will “sift” them by causing them to “pass under the rod,” and thus He will “purge out” those who do not have real faith (Ezek. 11:9-12; 20:34-38). A “tenth” of them (a remnant) will be found to be real and will be brought into “the bond of the covenant” (Isa. 6:13; Ezek. 20:37; Jer. 31:31-34). They will go up into the land and join the remnant of the two tribes (Jews) and they will be “one” and “shall be no more two nations, neither shall they be divided into two kingdoms any more at all” (Ezek. 37:14-22; Isa. 11:13; 49:18-23). This will be a triumphant moment because they have been separated for almost 3000 years! The mass of Israelites who will be purged out from among the ten tribes will be destroyed by the Lord Himself, when He roars out of Zion to destroy the last confederacy under Gog in the Winepress judgment (Rev. 14:17-20; Isa. 63:1-6; Joel 3:13-16). Two actions are mentioned in Revelation 14:19 in connection with Israel’s judgment. The “vine of the earth” (Israel) is first “gathered” (back to the land), and then the apostates among them are “cast” into the winepress of God’s judgment.
As mentioned, the word “remnant” is also used in connection with the Christian testimony (Rev. 2:24 – W. Kelly Trans.). It refers to those who are true believers amidst the mass of merely professing Christians. Today there are millions in the Christian profession, but the vast majority of these are only Christian nominally (in name only).