What is the second death?
The second death is the lake of fire, the final judgment where death and hell themselves are cast, and it is the eternal punishment for all whose names are not found in the book of life.
Scripture identifies the second death plainly in Revelation 20:14, which says, "And death and hell were cast into the lake of fire. This is the second death." This shows that the second death is not merely physical death, but the lake of fire itself, the final place of judgment where even death and hell as powers are ultimately thrown.
This judgment comes after the dead, both small and great, stand before God and are judged out of the books according to their works. Revelation 20:15 makes clear who faces this fate: "And whosoever was not found written in the book of life was cast into the lake of fire." So the second death is the eternal casting into the lake of fire of everyone whose name is missing from the book of life.
By contrast, those who have part in the first resurrection are shown to be completely free from this fate. Revelation 20:6 declares, "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." This shows that the second death has no claim or authority over those who belong to Christ through the first resurrection; its power and finality are reserved only for the wicked dead who are judged and found without their name in the book of life.
The scriptures
And death and hell were cast into the lake of fire. This is the second death.
And whosoever was not found written in the book of life was cast into the lake of fire.
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.
And I saw the dead, small and great, stand before God; and the books were opened: and another book was opened, which is the book of life: and the dead were judged out of those things which were written in the books, according to their works. And the sea gave up the dead which were in it; and death and hell delivered up the dead which were in them: and they were judged every man according to their works.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.