What is the rapture of the 144,000?
The rapture of the 144,000 refers to the taking up of these sealed Jewish male servants of God, who are shown standing with the Lamb on Mount Sion after being sealed and protected from the wrath that is about to fall on the earth.
The 144,000 are described as young Jewish men, virgins, who are innocent and serve the Lord. Before the wrath of God can be poured out on the earth, an angel ascends from the east with the seal of the living God and cries out to the four angels holding back the winds, telling them not to hurt the earth, the sea, or the trees until these servants of God are sealed in their foreheads. Twelve thousand are sealed from each of the tribes of Israel, making up the full number.
This sealing takes place during the events of the sixth seal, alongside the great earthquake, the sun turning black, the moon becoming as blood, and the stars falling to the earth. The order of events shown is that the 144,000 are sealed and raptured, and the Church is raptured as well, with the redeemed shouting salvation to God and the Lamb and falling before the throne in worship.
Revelation 14 gives the fuller picture of where they are taken. The Lamb stands on Mount Sion with the 144,000, who have his Father's name written in their foreheads. A voice like many waters and a great thunder is heard, and harpers are harping with their harps. They sing a new song before the throne, before the four beasts and the elders, a song that no one else can learn except the 144,000, because they alone were redeemed from the earth.
They are called the firstfruits unto God and to the Lamb, redeemed from among men. Scripture notes that they were not defiled with women, for they are virgins, and that they follow the Lamb wherever he goes. No guile was found in their mouths, and they are described as without fault before the throne of God. This is the rapture, or taking up, of the 144,000, set apart and preserved by God's seal before his wrath is poured out on the earth.
The scriptures
And after these things I saw four angels standing on the four corners of the earth, holding the four winds of the earth, that the wind should not blow on the earth, nor on the sea, nor on any tree. And I saw another angel ascending from the east, having the seal of the living God: and he cried with a loud voice to the four angels, to whom it was given to hurt the earth and the sea, Saying, Hurt not the earth, neither the sea, nor the trees, till we have sealed the servants of our God in their foreheads. And I heard the number of them which were sealed: and there were sealed an hundred and forty and four thousand of all the tribes of the children of Israel.
And I looked, and, lo, a Lamb stood on the mount Sion, and with him an hundred forty and four thousand, having his Father’s name written in their foreheads. And I heard a voice from heaven, as the voice of many waters, and as the voice of a great thunder: and I heard the voice of harpers harping with their harps: And they sung as it were a new song before the throne, and before the four beasts, and the elders: and no man could learn that song but the hundred and forty and four thousand, which were redeemed from the earth. These are they which were not defiled with women; for they are virgins. These are they which follow the Lamb whithersoever he goeth. These were redeemed from among men, being the firstfruits unto God and to the Lamb. And in their mouth was found no guile: for they are without fault before the throne of God.
And I beheld when he had opened the sixth seal, and, lo, there was a great earthquake; and the sun became black as sackcloth of hair, and the moon became as blood; And the stars of heaven fell unto the earth, even as a fig tree casteth her untimely figs, when she is shaken of a mighty wind. And the heaven departed as a scroll when it is rolled together; and every mountain and island were moved out of their places. And the kings of the earth, and the great men, and the rich men, and the chief captains, and the mighty men, and every bondman, and every free man, hid themselves in the dens and in the rocks of the mountains; And said to the mountains and rocks, Fall on us, and hide us from the face of him that sitteth on the throne, and from the wrath of the Lamb: For the great day of his wrath is come; and who shall be able to stand?
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.