The second death and the lake of fire occur in Revelation.
Revelation 2:11 He that hath an ear, let him hear what the Spirit saith unto the churches; He that overcometh shall not be hurt of the second death.
Revelation 20:6 Blessed and holy is he that hath part in the first resurrection: on such the second death hath no power, but they shall be priests of God and of Christ, and shall reign with him a thousand years.
Revelation 20:14 And death and hell (the grave) were cast into the lake of fire. This is the second death.
Revelation 21:8 But the fearful, and unbelieving, and the abominable, and murderers, and whoremongers, and sorcerers, and idolaters, and all liars, shall have their part in the lake which burneth with fire and brimstone: which is the second death.
The sentence upon Adam for his sin was death.
Genesis 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
Genesis 3:17… unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life; 18 Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field; 19 In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return.
The punishment for all sin is death.
Ezekiel 18:20 The soul that sinneth, it shall die…
Romans 6:23 For the wages of sin is death…
All sin and all die.
Romans 5:12 Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned:
In his Son, God has provided a way of escape from death. This way of escape was promised from the days of Adam.
Those who are ignorant of God die and perish like animals.
Psalms 49:20 Man that is in honour, and understandeth not, is like the beasts that perish.
Those who come to a knowledge of God and his purpose will be raised from the dead and judged on how they have responded.
Those who have been faithful will be given eternal life.
Those who have not been faithful will die again.
This is called the ‘second death’.
Revelation is a book of symbols.
The lake of fire is a symbol.
Death and hell (the grave) cannot be literally cast into a lake of fire.
The casting of them into the lake of fire symbolises that there will be no more death and no further need of graves.
Fire represents complete destruction.
From this we understand that those condemned to the second death perish forever without hope.
I hope you find this helpful.