'Eternal death' definitions:

Definition of 'Eternal death'

From: Easton

Eternal death The miserable fate of the wicked in hell (Matt. 25:46; Mark 3:29; Heb. 6:2; 2 Thess. 1:9; Matt. 18:8; 25:41; Jude 1:7). The Scripture as clearly teaches the unending duration of the penal sufferings of the lost as the "everlasting life," the "eternal life" of the righteous. The same Greek words in the New Testament (aion, aionios, aidios) are used to express (1) the eternal existence of God (1 Tim. 1:17; Rom. 1:20; 16:26); (2) of Christ (Rev. 1:18); (3) of the Holy Ghost (Heb. 9:14); and (4) the eternal duration of the sufferings of the lost (Matt. 25:46; Jude 1:6).
Their condition after casting off the mortal body is spoken of in these expressive words: "Fire that shall not be quenched" (Mark 9:45, 46), "fire unquenchable" (Luke 3:17), "the worm that never dies," the "bottomless pit" (Rev. 9:1), "the smoke of their torment ascending up for ever and ever" (Rev. 14:10, 11).
The idea that the "second death" (Rev. 20:14) is in the case of the wicked their absolute destruction, their annihilation, has not the slightest support from Scripture, which always represents their future as one of conscious suffering enduring for ever.
The supposition that God will ultimately secure the repentance and restoration of all sinners is equally unscriptural. There is not the slightest trace in all the Scriptures of any such restoration. Sufferings of themselves have no tendency to purify the soul from sin or impart spiritual life. The atoning death of Christ and the sanctifying power of the Holy Spirit are the only means of divine appointment for bringing men to repentance. Now in the case of them that perish these means have been rejected, and "there remaineth no more sacrifice for sins" (Heb. 10:26, 27).