First, notice the setting to which they refer—when Jesus "comes in His glory" (Matthew 25:31-32 Matthew 25:31-32  When the Son of man shall come in his glory, and all the holy angels with him, then shall he sit on the throne of his glory:
 And before him shall be gathered all nations: and he shall separate them one from another, as a shepherd divides his sheep from the goats:
American King James Version×). We are told that He separates the sheep from the goats. The sheep represent the righteous (Matthew 25:34-40 Matthew 25:34-40  Then shall the King say to them on his right hand, Come, you blessed of my Father, inherit the kingdom prepared for you from the foundation of the world:  For I was an hungered, and you gave me meat: I was thirsty, and you gave me drink: I was a stranger, and you took me in:  Naked, and you clothed me: I was sick, and you visited me: I was in prison, and you came to me.  Then shall the righteous answer him, saying, Lord, when saw we you an hungered, and fed you? or thirsty, and gave you drink?  When saw we you a stranger, and took you in? or naked, and clothed you?  Or when saw we you sick, or in prison, and came to you?  And the King shall answer and say to them, Truly I say to you, Inasmuch as you have done it to one of the least of these my brothers, you have done it to me.
American King James Version×). At His return He sets the sheep at His right hand. The goats in this instance represent sinners. They are appointed to assemble on Jesus' left hand. He then consigns the goats to "the everlasting fire prepared for the devil and his angels" (Matthew 25:41 Matthew 25:41Then shall he say also to them on the left hand, Depart from me, you cursed, into everlasting fire, prepared for the devil and his angels:
American King James Version×).
The word everlasting is translated from the Greek word aionios. The key to understanding this verse is knowing what will occur everlastingly. Does it refer to a fire that tortures without end, or does it have another meaning?
In Matthew 25:46 Matthew 25:46And these shall go away into everlasting punishment: but the righteous into life eternal.
American King James Version×Jesus spoke in a single sentence of everlasting (aionios) punishment and of life eternal (aionios). Since the righteous will be given eternal, or everlasting, life, many theologians believe the punishing of the wicked must last as long as the life given to the righteous. But this cannot be reconciled with the statement that those cast into the lake of fire perish—they are killed. As explained elsewhere, they suffer death—the second death (Revelation 2:11 Revelation 2:11He that has an ear, let him hear what the Spirit said to the churches; He that overcomes shall not be hurt of the second death.
American King James Version×; Revelation 20:6 Revelation 20:6Blessed and holy is he that has part in the first resurrection: on such the second death has no power, but they shall be priests of God and of Christ, and shall reign with him a thousand years.
American King James Version×, Revelation 14:1-20 Revelation 14:1-20  And I looked, and, see, a Lamb stood on the mount Sion, and with him an hundred forty and four thousand, having his Father's name written in their foreheads.  And I heard a voice from heaven, as the voice of many waters, and as the voice of a great thunder: and I heard the voice of harpers harping with their harps:  And they sung as it were a new song before the throne, and before the four beasts, and the elders: and no man could learn that song but the hundred and forty and four thousand, which were redeemed from the earth.  These are they which were not defiled with women; for they are virgins. These are they which follow the Lamb wherever he goes. These were redeemed from among men, being the first fruits to God and to the Lamb.  And in their mouth was found no guile: for they are without fault before the throne of God.  And I saw another angel fly in the middle of heaven, having the everlasting gospel to preach to them that dwell on the earth, and to every nation, and kindred, and tongue, and people,  Saying with a loud voice, Fear God, and give glory to him; for the hour of his judgment is come: and worship him that made heaven, and earth, and the sea, and the fountains of waters.  And there followed another angel, saying, Babylon is fallen, is fallen, that great city, because she made all nations drink of the wine of the wrath of her fornication.  And the third angel followed them, saying with a loud voice, If any man worship the beast and his image, and receive his mark in his forehead, or in his hand,  The same shall drink of the wine of the wrath of God, which is poured out without mixture into the cup of his indignation; and he shall be tormented with fire and brimstone in the presence of the holy angels, and in the presence of the Lamb:  And the smoke of their torment ascends up for ever and ever: and they have no rest day nor night, who worship the beast and his image, and whoever receives the mark of his name.  Here is the patience of the saints: here are they that keep the commandments of God, and the faith of Jesus.  And I heard a voice from heaven saying to me, Write, Blessed are the dead which die in the Lord from now on: Yes, said the Spirit, that they may rest from their labors; and their works do follow them.  And I looked, and behold a white cloud, and on the cloud one sat like to the Son of man, having on his head a golden crown, and in his hand a sharp sickle.  And another angel came out of the temple, crying with a loud voice to him that sat on the cloud, Thrust in your sickle, and reap: for the time is come for you to reap; for the harvest of the earth is ripe.  And he that sat on the cloud thrust in his sickle on the earth; and the earth was reaped.  And another angel came out of the temple which is in heaven, he also having a sharp sickle.  And another angel came out from the altar, which had power over fire; and cried with a loud cry to him that had the sharp sickle, saying, Thrust in your sharp sickle, and gather the clusters of the vine of the earth; for her grapes are fully ripe.  And the angel thrust in his sickle into the earth, and gathered the vine of the earth, and cast it into the great wine press of the wrath of God.  And the wine press was trodden without the city, and blood came out of the wine press, even to the horse bridles, by the space of a thousand and six hundred furlongs.
American King James Version×; Revelation 21:8 Revelation 21:8But the fearful, and unbelieving, and the abominable, and murderers, and fornicators, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burns with fire and brimstone: which is the second death.
American King James Version×).
A plain and simple meaning of Matthew 25:46 Matthew 25:46And these shall go away into everlasting punishment: but the righteous into life eternal.
American King James Version×that fits with the rest of the Bible is that the wicked are cast into a fire that annihilates them—renders them forever extinct. The resulting punishment of being cast into the aionios fire is a one-time event. It is a permanent punishment, the results of which will remain forever—that is, eternal death. It is not ongoing punishing that continues forever without end. This is the only explanation that agrees with the rest of the Scriptures.
An additional point needs to be made regarding the meaning of aionios. Genesis 19 describes God's destruction of two cities, Sodom and Gomorrah, for their wickedness: "Then the LORD rained brimstone and fire on Sodom and Gomorrah" (Genesis 19:24 Genesis 19:24Then the LORD rained on Sodom and on Gomorrah brimstone and fire from the LORD out of heaven;
American King James Version×). They were utterly destroyed—consumed by fire.
In the New Testament, the book of Jude describes these cities as "suffering the vengeance of eternal [aionios] fire" (Jude 1:7 Jude 1:7Even as Sodom and Gomorrha, and the cities about them in like manner, giving themselves over to fornication, and going after strange flesh, are set forth for an example, suffering the vengeance of eternal fire.
American King James Version×). Yet it is obvious that the fires that destroyed Sodom and Gomorrah are not still burning. In the case of these cities and in the case of the wicked, who are consigned to aionios fire, the fire burns and completely destroys. But the eternal aspect of the fire is its everlasting effect, not how long it actually burns.