The second death is the separation of the whole person from the presence and blessings of God forever. The second death involves not just the location separate from God, but, the condition experienced for all eternity apart from God: never-ending fire and brimstone, weeping and gnashing of teeth, suffering and torment. The location and the agonizing condition experienced in that location is called the second death.
The Bible does not teach annihilation – the idea that God obliterates unsaved people entirely from existence. Nor does the Bible teach soul sleep, an unconscious existence for eternity. The Bible says that people who are not saved will be resurrected (Rev. 20:12; John 5:28-29), judged, and heaved into the Lake of Fire – fully conscious, forever under the painful torments of this awful place.