What is the new heaven and the new earth?
The new heaven and new earth are the completely remade creation that God brings into being after the thousand year reign of Christ ends, once the wicked dead are judged and the old heaven and earth are purged by fire, so that a perfect, eternal paradise remains where God himself dwells with his people.
After the Millennium is finished, Satan is loosed one final time, gathers Gog and Magog against Jerusalem, and is destroyed by fire from heaven along with his armies. The devil is then cast into the lake of fire, and the great white throne judgment takes place, where the wicked dead are raised, judged out of the books, and cast into the lake of fire as the second death. It is at this point, once the old order has been purged, that God creates the new heaven and the new earth. Revelation 21:1 says, "And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea." The sea is removed because God no longer wants any division or separation among the nations or among his people.
This new creation is described as a place where God himself tabernacles with mankind forever. "Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God... And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away" (Revelation 21:3-4). In this new earth, John also sees a pure river of the water of life flowing from the throne of God and of the Lamb, with the tree of life on either side, bearing twelve kinds of fruit every month, its leaves for the healing of the nations (Revelation 22:1-2). Mankind is restored to speaking one pure language again, as Zephaniah 3:9 promised, so that all may call on the name of the Lord with one consent.
This final new heaven and earth is distinguished from the earthly renewal that happens at the start of the Millennium. Isaiah 65:17-20 speaks of a new heavens and earth where weeping is no more heard and life is greatly lengthened, but that describes the Millennium itself, the cleansing of the earth after the War of Armageddon. The new heaven and new earth of Revelation 21 and 22 come after that thousand year reign is over, once the current heavens and earth, kept in store and reserved unto fire, are dissolved. As 2 Peter 3:10-13 puts it, "the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up... Nevertheless we, according to his promise, look for new heavens and a new earth, wherein dwelleth righteousness."
So the new heaven and new earth is the eternal paradise that comes after all judgment is complete, a fully restored and even better creation than the original Eden, where sorrow, death, and pain are gone forever, and God dwells openly and permanently among his people.
The scriptures
And I saw a new heaven and a new earth: for the first heaven and the first earth were passed away; and there was no more sea. And I John saw the holy city, new Jerusalem, coming down from God out of heaven, prepared as a bride adorned for her husband. And I heard a great voice out of heaven saying, Behold, the tabernacle of God is with men, and he will dwell with them, and they shall be his people, and God himself shall be with them, and be their God. And God shall wipe away all tears from their eyes; and there shall be no more death, neither sorrow, nor crying, neither shall there be any more pain: for the former things are passed away.
And he shewed me a pure river of water of life, clear as crystal, proceeding out of the throne of God and of the Lamb. In the midst of the street of it, and on either side of the river, was there the tree of life, which bare twelve manner of fruits, and yielded her fruit every month: and the leaves of the tree were for the healing of the nations.
For then will I turn to the people a pure language, that they may all call upon the name of the LORD, to serve him with one consent.
For, behold, I create new heavens and a new earth: and the former shall not be remembered, nor come into mind. But be ye glad and rejoice for ever in that which I create: for, behold, I create Jerusalem a rejoicing, and her people a joy. And I will rejoice in Jerusalem, and joy in my people: and the voice of weeping shall be no more heard in her, nor the voice of crying. There shall be no more thence an infant of days, nor an old man that hath not filled his days: for the child shall die an hundred years old; but the sinner being an hundred years old shall be accursed.
But the heavens and the earth, which are now, by the same word are kept in store, reserved unto fire against the day of judgment and perdition of ungodly men. But, beloved, be not ignorant of this one thing, that one day is with the Lord as a thousand years, and a thousand years as one day. The Lord is not slack concerning his promise, as some men count slackness; but is longsuffering to us-ward, not willing that any should perish, but that all should come to repentance. But the day of the Lord will come as a thief in the night; in the which the heavens shall pass away with a great noise, and the elements shall melt with fervent heat, the earth also and the works that are therein shall be burned up. Seeing then that all these things shall be dissolved, what manner of persons ought ye to be in all holy conversation and godliness, Looking for and hasting unto the coming of the day of God, wherein the heavens being on fire shall be dissolved, and the elements shall melt with fervent heat? Nevertheless we, according to his promise, look for new heavens and a new earth, wherein dwelleth righteousness.
And I saw a great white throne, and him that sat on it, from whose face the earth and the heaven fled away; and there was found no place for them. 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. 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.
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.