Notes from Matthew 25:46
Eternal punishment takes place in hell (the lake of fire, or Gehenna), the place of punishment after death for all those who refuse to repent. In the Hebrew, three words were used in connection with eternal punishment.
(1) Sheol, or "the grave," was used in the Hebrew Old Testament to mean the place of the dead, generally thought to be under the earth. (In the Hebrew, this word occurs in Job 24:19; Psalm 16:10; Isaiah 38:10.)
(2) Hades is the Greek word for the underworld, the realm of the dead. It is the word used in the New Testament for Sheol. (In the Greek, this word occurs in Matthew 16:18; Revelation 1:18; 20:13, 14.)
(3) Gehenna, or hell, was named after the valley of Ben-Hinnom near Jerusalem, where children were sacrificed by fire to the pagan gods (see 2 Kings 23:10; 2 Chronicles 28:3). This is the place of eternal fire (Matthew 5:22; 10:28; Mark 9:43; Luke 12:5; James 3:6; Revelation 19:20) prepared for the devil, his angels, and all those who do not believe in God (25:46; Revelation 20:9, 10). This is the final and eternal state of the wicked after the resurrection and the Last Judgment.
When Jesus warns against unbelief, he is trying to save us from agonizing, eternal punishment.
Notes from 2 Thessalonians 1:9
The "eternal destruction" that Paul describes is the lake of fire (see Revelation 20:14)--the place of eternal separation from God. Those people who will be separated from God in eternity will no longer have any hope for salvation.
Notes from Jude 1:7
Many people don't want to believe that God sentences people to "eternal fire" for rejecting him. But this is clearly taught in Scripture. Sinners who don't seek forgiveness from God will face eternal separation from him. Jude warns all who rebel against, ignore, or reject God.